3.1.48 \(\int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx\) [48]

Optimal. Leaf size=274 \[ -\frac {3 a^4 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {9 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {b^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {6 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {4 a^3 b \csc (c+d x)}{d}-\frac {6 a b^3 \csc (c+d x)}{d}-\frac {3 a^4 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {4 a^3 b \csc ^3(c+d x)}{3 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {9 a^2 b^2 \sec (c+d x)}{d}+\frac {b^4 \sec (c+d x)}{d}-\frac {3 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{d}+\frac {2 a b^3 \csc (c+d x) \sec ^2(c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d} \]

[Out]

-3/8*a^4*arctanh(cos(d*x+c))/d-9*a^2*b^2*arctanh(cos(d*x+c))/d-b^4*arctanh(cos(d*x+c))/d+4*a^3*b*arctanh(sin(d
*x+c))/d+6*a*b^3*arctanh(sin(d*x+c))/d-4*a^3*b*csc(d*x+c)/d-6*a*b^3*csc(d*x+c)/d-3/8*a^4*cot(d*x+c)*csc(d*x+c)
/d-4/3*a^3*b*csc(d*x+c)^3/d-1/4*a^4*cot(d*x+c)*csc(d*x+c)^3/d+9*a^2*b^2*sec(d*x+c)/d+b^4*sec(d*x+c)/d-3*a^2*b^
2*csc(d*x+c)^2*sec(d*x+c)/d+2*a*b^3*csc(d*x+c)*sec(d*x+c)^2/d+1/3*b^4*sec(d*x+c)^3/d

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Rubi [A]
time = 0.19, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3598, 3853, 3855, 2701, 308, 213, 2702, 294, 327} \begin {gather*} -\frac {3 a^4 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3 a^4 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {4 a^3 b \csc ^3(c+d x)}{3 d}-\frac {4 a^3 b \csc (c+d x)}{d}+\frac {4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {9 a^2 b^2 \sec (c+d x)}{d}-\frac {9 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {3 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{d}-\frac {6 a b^3 \csc (c+d x)}{d}+\frac {6 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {2 a b^3 \csc (c+d x) \sec ^2(c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d}+\frac {b^4 \sec (c+d x)}{d}-\frac {b^4 \tanh ^{-1}(\cos (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^5*(a + b*Tan[c + d*x])^4,x]

[Out]

(-3*a^4*ArcTanh[Cos[c + d*x]])/(8*d) - (9*a^2*b^2*ArcTanh[Cos[c + d*x]])/d - (b^4*ArcTanh[Cos[c + d*x]])/d + (
4*a^3*b*ArcTanh[Sin[c + d*x]])/d + (6*a*b^3*ArcTanh[Sin[c + d*x]])/d - (4*a^3*b*Csc[c + d*x])/d - (6*a*b^3*Csc
[c + d*x])/d - (3*a^4*Cot[c + d*x]*Csc[c + d*x])/(8*d) - (4*a^3*b*Csc[c + d*x]^3)/(3*d) - (a^4*Cot[c + d*x]*Cs
c[c + d*x]^3)/(4*d) + (9*a^2*b^2*Sec[c + d*x])/d + (b^4*Sec[c + d*x])/d - (3*a^2*b^2*Csc[c + d*x]^2*Sec[c + d*
x])/d + (2*a*b^3*Csc[c + d*x]*Sec[c + d*x]^2)/d + (b^4*Sec[c + d*x]^3)/(3*d)

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2701

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 2702

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 3598

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[Expand[Sin[e
+ f*x]^m*(a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IGtQ[n, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \csc ^5(c+d x) (a+b \tan (c+d x))^4 \, dx &=\int \left (a^4 \csc ^5(c+d x)+4 a^3 b \csc ^4(c+d x) \sec (c+d x)+6 a^2 b^2 \csc ^3(c+d x) \sec ^2(c+d x)+4 a b^3 \csc ^2(c+d x) \sec ^3(c+d x)+b^4 \csc (c+d x) \sec ^4(c+d x)\right ) \, dx\\ &=a^4 \int \csc ^5(c+d x) \, dx+\left (4 a^3 b\right ) \int \csc ^4(c+d x) \sec (c+d x) \, dx+\left (6 a^2 b^2\right ) \int \csc ^3(c+d x) \sec ^2(c+d x) \, dx+\left (4 a b^3\right ) \int \csc ^2(c+d x) \sec ^3(c+d x) \, dx+b^4 \int \csc (c+d x) \sec ^4(c+d x) \, dx\\ &=-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{4} \left (3 a^4\right ) \int \csc ^3(c+d x) \, dx-\frac {\left (4 a^3 b\right ) \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (6 a^2 b^2\right ) \text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{d}-\frac {\left (4 a b^3\right ) \text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {b^4 \text {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {3 a^4 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{d}+\frac {2 a b^3 \csc (c+d x) \sec ^2(c+d x)}{d}+\frac {1}{8} \left (3 a^4\right ) \int \csc (c+d x) \, dx-\frac {\left (4 a^3 b\right ) \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (9 a^2 b^2\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}-\frac {\left (6 a b^3\right ) \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {b^4 \text {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {3 a^4 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {4 a^3 b \csc (c+d x)}{d}-\frac {6 a b^3 \csc (c+d x)}{d}-\frac {3 a^4 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {4 a^3 b \csc ^3(c+d x)}{3 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {9 a^2 b^2 \sec (c+d x)}{d}+\frac {b^4 \sec (c+d x)}{d}-\frac {3 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{d}+\frac {2 a b^3 \csc (c+d x) \sec ^2(c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d}-\frac {\left (4 a^3 b\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {\left (9 a^2 b^2\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}-\frac {\left (6 a b^3\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (c+d x)\right )}{d}+\frac {b^4 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {3 a^4 \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {9 a^2 b^2 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {b^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {4 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac {6 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {4 a^3 b \csc (c+d x)}{d}-\frac {6 a b^3 \csc (c+d x)}{d}-\frac {3 a^4 \cot (c+d x) \csc (c+d x)}{8 d}-\frac {4 a^3 b \csc ^3(c+d x)}{3 d}-\frac {a^4 \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {9 a^2 b^2 \sec (c+d x)}{d}+\frac {b^4 \sec (c+d x)}{d}-\frac {3 a^2 b^2 \csc ^2(c+d x) \sec (c+d x)}{d}+\frac {2 a b^3 \csc (c+d x) \sec ^2(c+d x)}{d}+\frac {b^4 \sec ^3(c+d x)}{3 d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1491\) vs. \(2(274)=548\).
time = 6.30, size = 1491, normalized size = 5.44 \begin {gather*} \frac {b^2 \left (36 a^2+7 b^2\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{6 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (-7 a^3 b \cos \left (\frac {1}{2} (c+d x)\right )-6 a b^3 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^4(c+d x) \csc \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{3 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {3 \left (a^4+8 a^2 b^2\right ) \cos ^4(c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{32 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {a^3 b \cos ^4(c+d x) \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{6 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {a^4 \cos ^4(c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{64 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (-3 a^4-72 a^2 b^2-8 b^4\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{8 d (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {2 \left (2 a^3 b+3 a b^3\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (3 a^4+72 a^2 b^2+8 b^4\right ) \cos ^4(c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{8 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {2 \left (2 a^3 b+3 a b^3\right ) \cos ^4(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {3 \left (a^4+8 a^2 b^2\right ) \cos ^4(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{32 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {a^4 \cos ^4(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{64 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (12 a b^3+b^4\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{12 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {b^4 \cos ^4(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {b^4 \cos ^4(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\left (-12 a b^3+b^4\right ) \cos ^4(c+d x) (a+b \tan (c+d x))^4}{12 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\cos ^4(c+d x) \sec \left (\frac {1}{2} (c+d x)\right ) \left (-7 a^3 b \sin \left (\frac {1}{2} (c+d x)\right )-6 a b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{3 d (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\cos ^4(c+d x) \left (-36 a^2 b^2 \sin \left (\frac {1}{2} (c+d x)\right )-7 b^4 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^4}+\frac {\cos ^4(c+d x) \left (36 a^2 b^2 \sin \left (\frac {1}{2} (c+d x)\right )+7 b^4 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^4}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^4}-\frac {a^3 b \cos ^4(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right ) (a+b \tan (c+d x))^4}{6 d (a \cos (c+d x)+b \sin (c+d x))^4} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Csc[c + d*x]^5*(a + b*Tan[c + d*x])^4,x]

[Out]

(b^2*(36*a^2 + 7*b^2)*Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(6*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-7*
a^3*b*Cos[(c + d*x)/2] - 6*a*b^3*Cos[(c + d*x)/2])*Cos[c + d*x]^4*Csc[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(3*
d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (3*(a^4 + 8*a^2*b^2)*Cos[c + d*x]^4*Csc[(c + d*x)/2]^2*(a + b*Tan[c +
d*x])^4)/(32*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (a^3*b*Cos[c + d*x]^4*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]
^2*(a + b*Tan[c + d*x])^4)/(6*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (a^4*Cos[c + d*x]^4*Csc[(c + d*x)/2]^4*
(a + b*Tan[c + d*x])^4)/(64*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-3*a^4 - 72*a^2*b^2 - 8*b^4)*Cos[c + d*
x]^4*Log[Cos[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (2*(2*a^3*b + 3
*a*b^3)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b
*Sin[c + d*x])^4) + ((3*a^4 + 72*a^2*b^2 + 8*b^4)*Cos[c + d*x]^4*Log[Sin[(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)
/(8*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (2*(2*a^3*b + 3*a*b^3)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] + Sin[
(c + d*x)/2]]*(a + b*Tan[c + d*x])^4)/(d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (3*(a^4 + 8*a^2*b^2)*Cos[c + d
*x]^4*Sec[(c + d*x)/2]^2*(a + b*Tan[c + d*x])^4)/(32*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (a^4*Cos[c + d*x
]^4*Sec[(c + d*x)/2]^4*(a + b*Tan[c + d*x])^4)/(64*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((12*a*b^3 + b^4)*
Cos[c + d*x]^4*(a + b*Tan[c + d*x])^4)/(12*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c
+ d*x])^4) + (b^4*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(6*d*(Cos[(c + d*x)/2] - Sin[(c + d
*x)/2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (b^4*Cos[c + d*x]^4*Sin[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/
(6*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + ((-12*a*b^3 + b^4)*Cos[c +
d*x]^4*(a + b*Tan[c + d*x])^4)/(12*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a*Cos[c + d*x] + b*Sin[c + d*x]
)^4) + (Cos[c + d*x]^4*Sec[(c + d*x)/2]*(-7*a^3*b*Sin[(c + d*x)/2] - 6*a*b^3*Sin[(c + d*x)/2])*(a + b*Tan[c +
d*x])^4)/(3*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) + (Cos[c + d*x]^4*(-36*a^2*b^2*Sin[(c + d*x)/2] - 7*b^4*Sin
[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(6*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c +
d*x])^4) + (Cos[c + d*x]^4*(36*a^2*b^2*Sin[(c + d*x)/2] + 7*b^4*Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^4)/(6*d
*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^4) - (a^3*b*Cos[c + d*x]^4*Sec[(c + d
*x)/2]^2*Tan[(c + d*x)/2]*(a + b*Tan[c + d*x])^4)/(6*d*(a*Cos[c + d*x] + b*Sin[c + d*x])^4)

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Maple [A]
time = 0.33, size = 241, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {a^{4} \left (\left (-\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+4 a^{3} b \left (-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+6 a^{2} b^{2} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+4 a \,b^{3} \left (\frac {1}{2 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+b^{4} \left (\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\)	\(241\)
default	\(\frac {a^{4} \left (\left (-\frac {\left (\csc ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \csc \left (d x +c \right )}{8}\right ) \cot \left (d x +c \right )+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8}\right )+4 a^{3} b \left (-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+6 a^{2} b^{2} \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+4 a \,b^{3} \left (\frac {1}{2 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {3}{2 \sin \left (d x +c \right )}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+b^{4} \left (\frac {1}{3 \cos \left (d x +c \right )^{3}}+\frac {1}{\cos \left (d x +c \right )}+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\)	\(241\)
risch	\(\frac {-216 a^{2} b^{2} {\mathrm e}^{9 i \left (d x +c \right )}+288 a^{2} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-216 a^{2} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-144 a^{2} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+216 a^{2} b^{2} {\mathrm e}^{i \left (d x +c \right )}-16 b^{4} {\mathrm e}^{11 i \left (d x +c \right )}-6 a^{4} {\mathrm e}^{11 i \left (d x +c \right )}+9 a^{4} {\mathrm e}^{13 i \left (d x +c \right )}+24 b^{4} {\mathrm e}^{13 i \left (d x +c \right )}+216 a^{2} b^{2} {\mathrm e}^{13 i \left (d x +c \right )}-144 a^{2} b^{2} {\mathrm e}^{11 i \left (d x +c \right )}-544 i a^{3} b \,{\mathrm e}^{5 i \left (d x +c \right )}+128 i a^{3} b \,{\mathrm e}^{11 i \left (d x +c \right )}-96 i a^{3} b \,{\mathrm e}^{13 i \left (d x +c \right )}+192 i a \,b^{3} {\mathrm e}^{11 i \left (d x +c \right )}-144 i a \,b^{3} {\mathrm e}^{13 i \left (d x +c \right )}-48 i a \,b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+96 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}-128 i a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+48 i a \,b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-192 i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+144 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+544 i a^{3} b \,{\mathrm e}^{9 i \left (d x +c \right )}-105 a^{4} {\mathrm e}^{9 i \left (d x +c \right )}-152 b^{4} {\mathrm e}^{9 i \left (d x +c \right )}-180 a^{4} {\mathrm e}^{7 i \left (d x +c \right )}+288 b^{4} {\mathrm e}^{7 i \left (d x +c \right )}-105 a^{4} {\mathrm e}^{5 i \left (d x +c \right )}-152 b^{4} {\mathrm e}^{5 i \left (d x +c \right )}-6 a^{4} {\mathrm e}^{3 i \left (d x +c \right )}-16 b^{4} {\mathrm e}^{3 i \left (d x +c \right )}+9 a^{4} {\mathrm e}^{i \left (d x +c \right )}+24 b^{4} {\mathrm e}^{i \left (d x +c \right )}}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {6 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {6 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {3 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d}-\frac {9 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{4}}{d}+\frac {3 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d}+\frac {9 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{4}}{d}\)	\(754\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^5*(a+b*tan(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^4*((-1/4*csc(d*x+c)^3-3/8*csc(d*x+c))*cot(d*x+c)+3/8*ln(csc(d*x+c)-cot(d*x+c)))+4*a^3*b*(-1/3/sin(d*x+c
)^3-1/sin(d*x+c)+ln(sec(d*x+c)+tan(d*x+c)))+6*a^2*b^2*(-1/2/sin(d*x+c)^2/cos(d*x+c)+3/2/cos(d*x+c)+3/2*ln(csc(
d*x+c)-cot(d*x+c)))+4*a*b^3*(1/2/sin(d*x+c)/cos(d*x+c)^2-3/2/sin(d*x+c)+3/2*ln(sec(d*x+c)+tan(d*x+c)))+b^4*(1/
3/cos(d*x+c)^3+1/cos(d*x+c)+ln(csc(d*x+c)-cot(d*x+c))))

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Maxima [A]
time = 0.34, size = 304, normalized size = 1.11 \begin {gather*} \frac {3 \, a^{4} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 72 \, a^{2} b^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 48 \, a b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{2} - 2\right )}}{\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 8 \, b^{4} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )}}{\cos \left (d x + c\right )^{3}} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 32 \, a^{3} b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{2} + 1\right )}}{\sin \left (d x + c\right )^{3}} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^5*(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

1/48*(3*a^4*(2*(3*cos(d*x + c)^3 - 5*cos(d*x + c))/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1) - 3*log(cos(d*x + c
) + 1) + 3*log(cos(d*x + c) - 1)) + 72*a^2*b^2*(2*(3*cos(d*x + c)^2 - 2)/(cos(d*x + c)^3 - cos(d*x + c)) - 3*l
og(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 48*a*b^3*(2*(3*sin(d*x + c)^2 - 2)/(sin(d*x + c)^3 - sin(d*x
+ c)) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) + 8*b^4*(2*(3*cos(d*x + c)^2 + 1)/cos(d*x + c)^3 -
3*log(cos(d*x + c) + 1) + 3*log(cos(d*x + c) - 1)) - 32*a^3*b*(2*(3*sin(d*x + c)^2 + 1)/sin(d*x + c)^3 - 3*lo
g(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (264) = 528\).
time = 0.47, size = 547, normalized size = 2.00 \begin {gather*} \frac {6 \, {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{6} - 10 \, {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 16 \, b^{4} + 16 \, {\left (18 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left ({\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{7} - 2 \, {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{5} + {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{7} - 2 \, {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{5} + {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 48 \, {\left ({\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{7} - 2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} + {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 48 \, {\left ({\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{7} - 2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} + {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 32 \, {\left (3 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} + 3 \, a b^{3} \cos \left (d x + c\right ) - 4 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{48 \, {\left (d \cos \left (d x + c\right )^{7} - 2 \, d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^5*(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/48*(6*(3*a^4 + 72*a^2*b^2 + 8*b^4)*cos(d*x + c)^6 - 10*(3*a^4 + 72*a^2*b^2 + 8*b^4)*cos(d*x + c)^4 + 16*b^4
+ 16*(18*a^2*b^2 + b^4)*cos(d*x + c)^2 - 3*((3*a^4 + 72*a^2*b^2 + 8*b^4)*cos(d*x + c)^7 - 2*(3*a^4 + 72*a^2*b^
2 + 8*b^4)*cos(d*x + c)^5 + (3*a^4 + 72*a^2*b^2 + 8*b^4)*cos(d*x + c)^3)*log(1/2*cos(d*x + c) + 1/2) + 3*((3*a
^4 + 72*a^2*b^2 + 8*b^4)*cos(d*x + c)^7 - 2*(3*a^4 + 72*a^2*b^2 + 8*b^4)*cos(d*x + c)^5 + (3*a^4 + 72*a^2*b^2
+ 8*b^4)*cos(d*x + c)^3)*log(-1/2*cos(d*x + c) + 1/2) + 48*((2*a^3*b + 3*a*b^3)*cos(d*x + c)^7 - 2*(2*a^3*b +
3*a*b^3)*cos(d*x + c)^5 + (2*a^3*b + 3*a*b^3)*cos(d*x + c)^3)*log(sin(d*x + c) + 1) - 48*((2*a^3*b + 3*a*b^3)*
cos(d*x + c)^7 - 2*(2*a^3*b + 3*a*b^3)*cos(d*x + c)^5 + (2*a^3*b + 3*a*b^3)*cos(d*x + c)^3)*log(-sin(d*x + c)
+ 1) + 32*(3*(2*a^3*b + 3*a*b^3)*cos(d*x + c)^5 + 3*a*b^3*cos(d*x + c) - 4*(2*a^3*b + 3*a*b^3)*cos(d*x + c)^3)
*sin(d*x + c))/(d*cos(d*x + c)^7 - 2*d*cos(d*x + c)^5 + d*cos(d*x + c)^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tan {\left (c + d x \right )}\right )^{4} \csc ^{5}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**5*(a+b*tan(d*x+c))**4,x)

[Out]

Integral((a + b*tan(c + d*x))**4*csc(c + d*x)**5, x)

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Giac [A]
time = 1.05, size = 479, normalized size = 1.75 \begin {gather*} \frac {3 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 32 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 144 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 480 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 384 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 384 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 384 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 24 \, {\left (3 \, a^{4} + 72 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {256 \, {\left (3 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 9 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 3 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 18 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 9 \, a^{2} b^{2} - 2 \, b^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}} - \frac {150 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3600 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 400 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 480 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 384 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 144 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 32 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{4}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^5*(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

1/192*(3*a^4*tan(1/2*d*x + 1/2*c)^4 - 32*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 24*a^4*tan(1/2*d*x + 1/2*c)^2 + 144*a^
2*b^2*tan(1/2*d*x + 1/2*c)^2 - 480*a^3*b*tan(1/2*d*x + 1/2*c) - 384*a*b^3*tan(1/2*d*x + 1/2*c) + 384*(2*a^3*b
+ 3*a*b^3)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 384*(2*a^3*b + 3*a*b^3)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2
4*(3*a^4 + 72*a^2*b^2 + 8*b^4)*log(abs(tan(1/2*d*x + 1/2*c))) + 256*(3*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 9*a^2*b^
2*tan(1/2*d*x + 1/2*c)^4 - 3*b^4*tan(1/2*d*x + 1/2*c)^4 + 18*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 3*b^4*tan(1/2*d*
x + 1/2*c)^2 - 3*a*b^3*tan(1/2*d*x + 1/2*c) - 9*a^2*b^2 - 2*b^4)/(tan(1/2*d*x + 1/2*c)^2 - 1)^3 - (150*a^4*tan
(1/2*d*x + 1/2*c)^4 + 3600*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 400*b^4*tan(1/2*d*x + 1/2*c)^4 + 480*a^3*b*tan(1/2
*d*x + 1/2*c)^3 + 384*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 24*a^4*tan(1/2*d*x + 1/2*c)^2 + 144*a^2*b^2*tan(1/2*d*x +
1/2*c)^2 + 32*a^3*b*tan(1/2*d*x + 1/2*c) + 3*a^4)/tan(1/2*d*x + 1/2*c)^4)/d

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Mupad [B]
time = 4.17, size = 857, normalized size = 3.13 \begin {gather*} \frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {\mathrm {atan}\left (-\frac {\left (4\,a^3\,b+6\,a\,b^3\right )\,\left (12\,a\,b^3+8\,a^3\,b-6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,a^3\,b+6\,a\,b^3\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^4}{4}+18\,a^2\,b^2+2\,b^4\right )\right )\,1{}\mathrm {i}+\left (4\,a^3\,b+6\,a\,b^3\right )\,\left (12\,a\,b^3+8\,a^3\,b+6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,a^3\,b+6\,a\,b^3\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^4}{4}+18\,a^2\,b^2+2\,b^4\right )\right )\,1{}\mathrm {i}}{2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (64\,a^6\,b^2+192\,a^4\,b^4+144\,a^2\,b^6\right )+\left (4\,a^3\,b+6\,a\,b^3\right )\,\left (12\,a\,b^3+8\,a^3\,b-6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,a^3\,b+6\,a\,b^3\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^4}{4}+18\,a^2\,b^2+2\,b^4\right )\right )-\left (4\,a^3\,b+6\,a\,b^3\right )\,\left (12\,a\,b^3+8\,a^3\,b+6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (4\,a^3\,b+6\,a\,b^3\right )-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^4}{4}+18\,a^2\,b^2+2\,b^4\right )\right )+24\,a\,b^7+6\,a^7\,b+232\,a^3\,b^5+153\,a^5\,b^3}\right )\,\left (a^3\,b\,8{}\mathrm {i}+a\,b^3\,12{}\mathrm {i}\right )}{d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^3\,b+\frac {a\,b\,\left (a^2+4\,b^2\right )}{2}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^4}{8}+9\,a^2\,b^2+b^4\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^4}{8}+\frac {3\,a^2\,b^2}{4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {23\,a^4}{4}+420\,a^2\,b^2+64\,b^4\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {21\,a^4}{4}+228\,a^2\,b^2+\frac {128\,b^4}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (2\,a^4+204\,a^2\,b^2+64\,b^4\right )+\frac {a^4}{4}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {5\,a^4}{4}+12\,a^2\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (32\,a^3\,b+32\,a\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (32\,a\,b^3-40\,a^3\,b\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (112\,a^3\,b+160\,a\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {352\,a^3\,b}{3}+96\,a\,b^3\right )+\frac {8\,a^3\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}}{d\,\left (-16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}-\frac {a^3\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tan(c + d*x))^4/sin(c + d*x)^5,x)

[Out]

(a^4*tan(c/2 + (d*x)/2)^4)/(64*d) - (atan(-((6*a*b^3 + 4*a^3*b)*(12*a*b^3 + 8*a^3*b - 6*tan(c/2 + (d*x)/2)*(6*
a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((3*a^4)/4 + 2*b^4 + 18*a^2*b^2))*1i + (6*a*b^3 + 4*a^3*b)*(12*a*b^3 + 8
*a^3*b + 6*tan(c/2 + (d*x)/2)*(6*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((3*a^4)/4 + 2*b^4 + 18*a^2*b^2))*1i)/(
2*tan(c/2 + (d*x)/2)*(144*a^2*b^6 + 192*a^4*b^4 + 64*a^6*b^2) + (6*a*b^3 + 4*a^3*b)*(12*a*b^3 + 8*a^3*b - 6*ta
n(c/2 + (d*x)/2)*(6*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((3*a^4)/4 + 2*b^4 + 18*a^2*b^2)) - (6*a*b^3 + 4*a^3
*b)*(12*a*b^3 + 8*a^3*b + 6*tan(c/2 + (d*x)/2)*(6*a*b^3 + 4*a^3*b) - tan(c/2 + (d*x)/2)*((3*a^4)/4 + 2*b^4 + 1
8*a^2*b^2)) + 24*a*b^7 + 6*a^7*b + 232*a^3*b^5 + 153*a^5*b^3))*(a*b^3*12i + a^3*b*8i))/d - (tan(c/2 + (d*x)/2)
*(2*a^3*b + (a*b*(a^2 + 4*b^2))/2))/d + (log(tan(c/2 + (d*x)/2))*((3*a^4)/8 + b^4 + 9*a^2*b^2))/d + (tan(c/2 +
(d*x)/2)^2*(a^4/8 + (3*a^2*b^2)/4))/d - (tan(c/2 + (d*x)/2)^6*((23*a^4)/4 + 64*b^4 + 420*a^2*b^2) - tan(c/2 +
(d*x)/2)^4*((21*a^4)/4 + (128*b^4)/3 + 228*a^2*b^2) - tan(c/2 + (d*x)/2)^8*(2*a^4 + 64*b^4 + 204*a^2*b^2) + a
^4/4 + tan(c/2 + (d*x)/2)^2*((5*a^4)/4 + 12*a^2*b^2) + tan(c/2 + (d*x)/2)^3*(32*a*b^3 + 32*a^3*b) + tan(c/2 +
(d*x)/2)^9*(32*a*b^3 - 40*a^3*b) - tan(c/2 + (d*x)/2)^5*(160*a*b^3 + 112*a^3*b) + tan(c/2 + (d*x)/2)^7*(96*a*b
^3 + (352*a^3*b)/3) + (8*a^3*b*tan(c/2 + (d*x)/2))/3)/(d*(16*tan(c/2 + (d*x)/2)^4 - 48*tan(c/2 + (d*x)/2)^6 +
48*tan(c/2 + (d*x)/2)^8 - 16*tan(c/2 + (d*x)/2)^10)) - (a^3*b*tan(c/2 + (d*x)/2)^3)/(6*d)

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