\(\int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx\) [103]

Optimal result

Integrand size = 21, antiderivative size = 257 \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b d^{5/2}}+\frac {3 \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b d^{5/2}}-\frac {3 \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b d^{5/2}}+\frac {3 \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b d^{5/2}}+\frac {\cos ^2(a+b x) \sqrt {d \tan (a+b x)}}{16 b d^3}-\frac {\cos ^4(a+b x) \sqrt {d \tan (a+b x)}}{4 b d^3} \]

[Out]

-3/64*arctan(1-2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2))/b/d^(5/2)*2^(1/2)+3/64*arctan(1+2^(1/2)*(d*tan(b*x+a))^(1
/2)/d^(1/2))/b/d^(5/2)*2^(1/2)-3/128*ln(d^(1/2)-2^(1/2)*(d*tan(b*x+a))^(1/2)+d^(1/2)*tan(b*x+a))/b/d^(5/2)*2^(
1/2)+3/128*ln(d^(1/2)+2^(1/2)*(d*tan(b*x+a))^(1/2)+d^(1/2)*tan(b*x+a))/b/d^(5/2)*2^(1/2)+1/16*cos(b*x+a)^2*(d*
tan(b*x+a))^(1/2)/b/d^3-1/4*cos(b*x+a)^4*(d*tan(b*x+a))^(1/2)/b/d^3

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2671, 294, 296, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b d^{5/2}}+\frac {3 \arctan \left (\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}+1\right )}{32 \sqrt {2} b d^{5/2}}-\frac {3 \log \left (\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}+\sqrt {d}\right )}{64 \sqrt {2} b d^{5/2}}+\frac {3 \log \left (\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}+\sqrt {d}\right )}{64 \sqrt {2} b d^{5/2}}-\frac {\cos ^4(a+b x) \sqrt {d \tan (a+b x)}}{4 b d^3}+\frac {\cos ^2(a+b x) \sqrt {d \tan (a+b x)}}{16 b d^3} \]

[In]

Int[Sin[a + b*x]^4/(d*Tan[a + b*x])^(5/2),x]

[Out]

(-3*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[a + b*x]])/Sqrt[d]])/(32*Sqrt[2]*b*d^(5/2)) + (3*ArcTan[1 + (Sqrt[2]*Sqrt[d
*Tan[a + b*x]])/Sqrt[d]])/(32*Sqrt[2]*b*d^(5/2)) - (3*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] - Sqrt[2]*Sqrt[d*Tan[
a + b*x]]])/(64*Sqrt[2]*b*d^(5/2)) + (3*Log[Sqrt[d] + Sqrt[d]*Tan[a + b*x] + Sqrt[2]*Sqrt[d*Tan[a + b*x]]])/(6
4*Sqrt[2]*b*d^(5/2)) + (Cos[a + b*x]^2*Sqrt[d*Tan[a + b*x]])/(16*b*d^3) - (Cos[a + b*x]^4*Sqrt[d*Tan[a + b*x]]
)/(4*b*d^3)

Rule 210

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

Rule 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

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 296

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

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 2671

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> With[{ff = FreeFactors[Ta
n[e + f*x], x]}, Dist[b*(ff/f), Subst[Int[(ff*x)^(m + n)/(b^2 + ff^2*x^2)^(m/2 + 1), x], x, b*(Tan[e + f*x]/ff
)], x]] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2]

Rubi steps \begin{align*} \text {integral}& = \frac {d \text {Subst}\left (\int \frac {x^{3/2}}{\left (d^2+x^2\right )^3} \, dx,x,d \tan (a+b x)\right )}{b} \\ & = -\frac {\cos ^4(a+b x) \sqrt {d \tan (a+b x)}}{4 b d^3}+\frac {d \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (d^2+x^2\right )^2} \, dx,x,d \tan (a+b x)\right )}{8 b} \\ & = \frac {\cos ^2(a+b x) \sqrt {d \tan (a+b x)}}{16 b d^3}-\frac {\cos ^4(a+b x) \sqrt {d \tan (a+b x)}}{4 b d^3}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (d^2+x^2\right )} \, dx,x,d \tan (a+b x)\right )}{32 b d} \\ & = \frac {\cos ^2(a+b x) \sqrt {d \tan (a+b x)}}{16 b d^3}-\frac {\cos ^4(a+b x) \sqrt {d \tan (a+b x)}}{4 b d^3}+\frac {3 \text {Subst}\left (\int \frac {1}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{16 b d} \\ & = \frac {\cos ^2(a+b x) \sqrt {d \tan (a+b x)}}{16 b d^3}-\frac {\cos ^4(a+b x) \sqrt {d \tan (a+b x)}}{4 b d^3}+\frac {3 \text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{32 b d^2}+\frac {3 \text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{32 b d^2} \\ & = \frac {\cos ^2(a+b x) \sqrt {d \tan (a+b x)}}{16 b d^3}-\frac {\cos ^4(a+b x) \sqrt {d \tan (a+b x)}}{4 b d^3}-\frac {3 \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b d^{5/2}}-\frac {3 \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b d^{5/2}}+\frac {3 \text {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{64 b d^2}+\frac {3 \text {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (a+b x)}\right )}{64 b d^2} \\ & = -\frac {3 \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b d^{5/2}}+\frac {3 \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b d^{5/2}}+\frac {\cos ^2(a+b x) \sqrt {d \tan (a+b x)}}{16 b d^3}-\frac {\cos ^4(a+b x) \sqrt {d \tan (a+b x)}}{4 b d^3}+\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b d^{5/2}}-\frac {3 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b d^{5/2}} \\ & = -\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b d^{5/2}}+\frac {3 \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{32 \sqrt {2} b d^{5/2}}-\frac {3 \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b d^{5/2}}+\frac {3 \log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{64 \sqrt {2} b d^{5/2}}+\frac {\cos ^2(a+b x) \sqrt {d \tan (a+b x)}}{16 b d^3}-\frac {\cos ^4(a+b x) \sqrt {d \tan (a+b x)}}{4 b d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.48 \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=-\frac {\csc (a+b x) \left (\sin (a+b x)+3 \arcsin (\cos (a+b x)-\sin (a+b x)) \sqrt {\sin (2 (a+b x))}-3 \log \left (\cos (a+b x)+\sin (a+b x)+\sqrt {\sin (2 (a+b x))}\right ) \sqrt {\sin (2 (a+b x))}+2 \sin (3 (a+b x))+\sin (5 (a+b x))\right ) \sqrt {d \tan (a+b x)}}{64 b d^3} \]

[In]

Integrate[Sin[a + b*x]^4/(d*Tan[a + b*x])^(5/2),x]

[Out]

-1/64*(Csc[a + b*x]*(Sin[a + b*x] + 3*ArcSin[Cos[a + b*x] - Sin[a + b*x]]*Sqrt[Sin[2*(a + b*x)]] - 3*Log[Cos[a
+ b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]*Sqrt[Sin[2*(a + b*x)]] + 2*Sin[3*(a + b*x)] + Sin[5*(a + b*x)
])*Sqrt[d*Tan[a + b*x]])/(b*d^3)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(604\) vs. \(2(197)=394\).

Time = 15.09 (sec) , antiderivative size = 605, normalized size of antiderivative = 2.35

method	result	size
default	\(\frac {\csc \left (b x +a \right ) \left (-1+\cos \left (b x +a \right )\right ) \left (16 \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {2}\, \left (\cos ^{4}\left (b x +a \right )\right )+16 \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {2}\, \left (\cos ^{3}\left (b x +a \right )\right )-4 \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {2}\, \left (\cos ^{2}\left (b x +a \right )\right )-4 \cos \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+6 \arctan \left (\frac {\sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}+\cos \left (b x +a \right )-1}{-1+\cos \left (b x +a \right )}\right )+6 \arctan \left (\frac {\sin \left (b x +a \right ) \sqrt {2}\, \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}-\cos \left (b x +a \right )+1}{-1+\cos \left (b x +a \right )}\right )-3 \ln \left (\frac {2 \sin \left (b x +a \right ) \sqrt {-\left (\cot ^{3}\left (b x +a \right )\right )+3 \left (\cot ^{2}\left (b x +a \right )\right ) \csc \left (b x +a \right )-3 \cot \left (b x +a \right ) \left (\csc ^{2}\left (b x +a \right )\right )+\csc ^{3}\left (b x +a \right )+\cot \left (b x +a \right )-\csc \left (b x +a \right )}-\cot \left (b x +a \right ) \cos \left (b x +a \right )+2 \cot \left (b x +a \right )+2 \cos \left (b x +a \right )+\sin \left (b x +a \right )-\csc \left (b x +a \right )-2}{-1+\cos \left (b x +a \right )}\right )+3 \ln \left (-\frac {\cot \left (b x +a \right ) \cos \left (b x +a \right )-2 \cot \left (b x +a \right )+2 \sin \left (b x +a \right ) \sqrt {-\left (\cot ^{3}\left (b x +a \right )\right )+3 \left (\cot ^{2}\left (b x +a \right )\right ) \csc \left (b x +a \right )-3 \cot \left (b x +a \right ) \left (\csc ^{2}\left (b x +a \right )\right )+\csc ^{3}\left (b x +a \right )+\cot \left (b x +a \right )-\csc \left (b x +a \right )}-2 \cos \left (b x +a \right )-\sin \left (b x +a \right )+\csc \left (b x +a \right )+2}{-1+\cos \left (b x +a \right )}\right )\right ) \sqrt {2}}{128 b \sqrt {-\frac {\cos \left (b x +a \right ) \sin \left (b x +a \right )}{\left (\cos \left (b x +a \right )+1\right )^{2}}}\, \sqrt {d \tan \left (b x +a \right )}\, d^{2}}\)	\(605\)

[In]

int(sin(b*x+a)^4/(d*tan(b*x+a))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/128/b*csc(b*x+a)*(-1+cos(b*x+a))*(16*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*2^(1/2)*cos(b*x+a)^4+16
*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)*2^(1/2)*cos(b*x+a)^3-4*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)
^2)^(1/2)*2^(1/2)*cos(b*x+a)^2-4*cos(b*x+a)*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+6*arctan((
sin(b*x+a)*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)+cos(b*x+a)-1)/(-1+cos(b*x+a)))+6*arctan((si
n(b*x+a)*2^(1/2)*(-cos(b*x+a)*sin(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)-cos(b*x+a)+1)/(-1+cos(b*x+a)))-3*ln((2*sin(b*
x+a)*(-cot(b*x+a)^3+3*cot(b*x+a)^2*csc(b*x+a)-3*cot(b*x+a)*csc(b*x+a)^2+csc(b*x+a)^3+cot(b*x+a)-csc(b*x+a))^(1
/2)-cot(b*x+a)*cos(b*x+a)+2*cot(b*x+a)+2*cos(b*x+a)+sin(b*x+a)-csc(b*x+a)-2)/(-1+cos(b*x+a)))+3*ln(-(cot(b*x+a
)*cos(b*x+a)-2*cot(b*x+a)+2*sin(b*x+a)*(-cot(b*x+a)^3+3*cot(b*x+a)^2*csc(b*x+a)-3*cot(b*x+a)*csc(b*x+a)^2+csc(
b*x+a)^3+cot(b*x+a)-csc(b*x+a))^(1/2)-2*cos(b*x+a)-sin(b*x+a)+csc(b*x+a)+2)/(-1+cos(b*x+a))))/(-cos(b*x+a)*sin
(b*x+a)/(cos(b*x+a)+1)^2)^(1/2)/(d*tan(b*x+a))^(1/2)/d^2*2^(1/2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 956, normalized size of antiderivative = 3.72 \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(sin(b*x+a)^4/(d*tan(b*x+a))^(5/2),x, algorithm="fricas")

[Out]

1/256*(3*b*d^3*(-1/(b^4*d^10))^(1/4)*log(2*b^2*d^5*sqrt(-1/(b^4*d^10))*cos(b*x + a)*sin(b*x + a) - 2*cos(b*x +
a)^2 + 2*(b^3*d^7*(-1/(b^4*d^10))^(3/4)*cos(b*x + a)^2 + b*d^2*(-1/(b^4*d^10))^(1/4)*cos(b*x + a)*sin(b*x + a
))*sqrt(d*sin(b*x + a)/cos(b*x + a)) + 1) - 3*b*d^3*(-1/(b^4*d^10))^(1/4)*log(2*b^2*d^5*sqrt(-1/(b^4*d^10))*co
s(b*x + a)*sin(b*x + a) - 2*cos(b*x + a)^2 - 2*(b^3*d^7*(-1/(b^4*d^10))^(3/4)*cos(b*x + a)^2 + b*d^2*(-1/(b^4*
d^10))^(1/4)*cos(b*x + a)*sin(b*x + a))*sqrt(d*sin(b*x + a)/cos(b*x + a)) + 1) + 3*I*b*d^3*(-1/(b^4*d^10))^(1/
4)*log(-2*b^2*d^5*sqrt(-1/(b^4*d^10))*cos(b*x + a)*sin(b*x + a) - 2*cos(b*x + a)^2 - 2*(I*b^3*d^7*(-1/(b^4*d^1
0))^(3/4)*cos(b*x + a)^2 - I*b*d^2*(-1/(b^4*d^10))^(1/4)*cos(b*x + a)*sin(b*x + a))*sqrt(d*sin(b*x + a)/cos(b*
x + a)) + 1) - 3*I*b*d^3*(-1/(b^4*d^10))^(1/4)*log(-2*b^2*d^5*sqrt(-1/(b^4*d^10))*cos(b*x + a)*sin(b*x + a) -
2*cos(b*x + a)^2 - 2*(-I*b^3*d^7*(-1/(b^4*d^10))^(3/4)*cos(b*x + a)^2 + I*b*d^2*(-1/(b^4*d^10))^(1/4)*cos(b*x
+ a)*sin(b*x + a))*sqrt(d*sin(b*x + a)/cos(b*x + a)) + 1) + 3*b*d^3*(-1/(b^4*d^10))^(1/4)*log(2*(b^3*d^7*(-1/(
b^4*d^10))^(3/4)*cos(b*x + a)^2 - b*d^2*(-1/(b^4*d^10))^(1/4)*cos(b*x + a)*sin(b*x + a))*sqrt(d*sin(b*x + a)/c
os(b*x + a)) - 1) - 3*b*d^3*(-1/(b^4*d^10))^(1/4)*log(-2*(b^3*d^7*(-1/(b^4*d^10))^(3/4)*cos(b*x + a)^2 - b*d^2
*(-1/(b^4*d^10))^(1/4)*cos(b*x + a)*sin(b*x + a))*sqrt(d*sin(b*x + a)/cos(b*x + a)) - 1) + 3*I*b*d^3*(-1/(b^4*
d^10))^(1/4)*log(-2*(I*b^3*d^7*(-1/(b^4*d^10))^(3/4)*cos(b*x + a)^2 + I*b*d^2*(-1/(b^4*d^10))^(1/4)*cos(b*x +
a)*sin(b*x + a))*sqrt(d*sin(b*x + a)/cos(b*x + a)) - 1) - 3*I*b*d^3*(-1/(b^4*d^10))^(1/4)*log(-2*(-I*b^3*d^7*(
-1/(b^4*d^10))^(3/4)*cos(b*x + a)^2 - I*b*d^2*(-1/(b^4*d^10))^(1/4)*cos(b*x + a)*sin(b*x + a))*sqrt(d*sin(b*x
+ a)/cos(b*x + a)) - 1) - 16*(4*cos(b*x + a)^4 - cos(b*x + a)^2)*sqrt(d*sin(b*x + a)/cos(b*x + a)))/(b*d^3)

Sympy [F]

\[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\int \frac {\sin ^{4}{\left (a + b x \right )}}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(sin(b*x+a)**4/(d*tan(b*x+a))**(5/2),x)

[Out]

Integral(sin(a + b*x)**4/(d*tan(a + b*x))**(5/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.85 \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\frac {6 \, \sqrt {2} d^{\frac {5}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right ) + 6 \, \sqrt {2} d^{\frac {5}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right ) + 3 \, \sqrt {2} d^{\frac {5}{2}} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right ) - 3 \, \sqrt {2} d^{\frac {5}{2}} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right ) + \frac {8 \, {\left (\left (d \tan \left (b x + a\right )\right )^{\frac {5}{2}} d^{4} - 3 \, \sqrt {d \tan \left (b x + a\right )} d^{6}\right )}}{d^{4} \tan \left (b x + a\right )^{4} + 2 \, d^{4} \tan \left (b x + a\right )^{2} + d^{4}}}{128 \, b d^{5}} \]

[In]

integrate(sin(b*x+a)^4/(d*tan(b*x+a))^(5/2),x, algorithm="maxima")

[Out]

1/128*(6*sqrt(2)*d^(5/2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d*tan(b*x + a)))/sqrt(d)) + 6*sqrt(2)*d^
(5/2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(d) - 2*sqrt(d*tan(b*x + a)))/sqrt(d)) + 3*sqrt(2)*d^(5/2)*log(d*tan(b*
x + a) + sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(d) + d) - 3*sqrt(2)*d^(5/2)*log(d*tan(b*x + a) - sqrt(2)*sqrt(d*tan
(b*x + a))*sqrt(d) + d) + 8*((d*tan(b*x + a))^(5/2)*d^4 - 3*sqrt(d*tan(b*x + a))*d^6)/(d^4*tan(b*x + a)^4 + 2*
d^4*tan(b*x + a)^2 + d^4))/(b*d^5)

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.96 \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\frac {3 \, \sqrt {2} \sqrt {{\left | d \right |}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{64 \, b d^{3}} + \frac {3 \, \sqrt {2} \sqrt {{\left | d \right |}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{64 \, b d^{3}} + \frac {3 \, \sqrt {2} \sqrt {{\left | d \right |}} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{128 \, b d^{3}} - \frac {3 \, \sqrt {2} \sqrt {{\left | d \right |}} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{128 \, b d^{3}} + \frac {\sqrt {d \tan \left (b x + a\right )} d^{2} \tan \left (b x + a\right )^{2} - 3 \, \sqrt {d \tan \left (b x + a\right )} d^{2}}{16 \, {\left (d^{2} \tan \left (b x + a\right )^{2} + d^{2}\right )}^{2} b d} \]

[In]

integrate(sin(b*x+a)^4/(d*tan(b*x+a))^(5/2),x, algorithm="giac")

[Out]

3/64*sqrt(2)*sqrt(abs(d))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) + 2*sqrt(d*tan(b*x + a)))/sqrt(abs(d)))/(b*
d^3) + 3/64*sqrt(2)*sqrt(abs(d))*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) - 2*sqrt(d*tan(b*x + a)))/sqrt(abs(
d)))/(b*d^3) + 3/128*sqrt(2)*sqrt(abs(d))*log(d*tan(b*x + a) + sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(abs(d)) + abs
(d))/(b*d^3) - 3/128*sqrt(2)*sqrt(abs(d))*log(d*tan(b*x + a) - sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(abs(d)) + abs
(d))/(b*d^3) + 1/16*(sqrt(d*tan(b*x + a))*d^2*tan(b*x + a)^2 - 3*sqrt(d*tan(b*x + a))*d^2)/((d^2*tan(b*x + a)^
2 + d^2)^2*b*d)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin ^4(a+b x)}{(d \tan (a+b x))^{5/2}} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^4}{{\left (d\,\mathrm {tan}\left (a+b\,x\right )\right )}^{5/2}} \,d x \]

[In]

int(sin(a + b*x)^4/(d*tan(a + b*x))^(5/2),x)

[Out]

int(sin(a + b*x)^4/(d*tan(a + b*x))^(5/2), x)