3.5.89 \(\int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^4} \, dx\) [489]

Optimal. Leaf size=189 \[ -\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (a^2+7 b^2\right )}{6 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))} \]

[Out]

-4*a*b*(a^2-b^2)*x/(a^2+b^2)^4+(a^4-6*a^2*b^2+b^4)*ln(a*cos(d*x+c)+b*sin(d*x+c))/(a^2+b^2)^4/d-1/3*a^2*tan(d*x
+c)/b/(a^2+b^2)/d/(a+b*tan(d*x+c))^3-1/6*a^2*(a^2+7*b^2)/b^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^2-a*(a^2-3*b^2)/(a
^2+b^2)^3/d/(a+b*tan(d*x+c))

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Rubi [A]
time = 0.23, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3646, 3709, 3610, 3612, 3611} \begin {gather*} -\frac {a^2 \left (a^2+7 b^2\right )}{6 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac {a \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}-\frac {4 a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^4}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^3/(a + b*Tan[c + d*x])^4,x]

[Out]

(-4*a*b*(a^2 - b^2)*x)/(a^2 + b^2)^4 + ((a^4 - 6*a^2*b^2 + b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 +
b^2)^4*d) - (a^2*Tan[c + d*x])/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^3) - (a^2*(a^2 + 7*b^2))/(6*b^2*(a^2 +
b^2)^2*d*(a + b*Tan[c + d*x])^2) - (a*(a^2 - 3*b^2))/((a^2 + b^2)^3*d*(a + b*Tan[c + d*x]))

Rule 3610

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b
*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3611

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

Rule 3612

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c +
b*d)*(x/(a^2 + b^2)), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rule 3646

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3709

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2)
)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e +
f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2
+ b^2, 0]

Rubi steps

\begin {align*} \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^4} \, dx &=-\frac {a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}+\frac {\int \frac {a^2-3 a b \tan (c+d x)+\left (a^2+3 b^2\right ) \tan ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx}{3 b \left (a^2+b^2\right )}\\ &=-\frac {a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (a^2+7 b^2\right )}{6 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\int \frac {-6 a b^2-3 b \left (a^2-b^2\right ) \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx}{3 b \left (a^2+b^2\right )^2}\\ &=-\frac {a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (a^2+7 b^2\right )}{6 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\int \frac {-3 b^2 \left (3 a^2-b^2\right )-3 a b \left (a^2-3 b^2\right ) \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{3 b \left (a^2+b^2\right )^3}\\ &=-\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}-\frac {a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (a^2+7 b^2\right )}{6 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\left (a^2+b^2\right )^4}\\ &=-\frac {4 a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^4}+\frac {\left (a^4-6 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac {a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^3}-\frac {a^2 \left (a^2+7 b^2\right )}{6 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}-\frac {a \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 6.24, size = 387, normalized size = 2.05 \begin {gather*} -\frac {\tan (c+d x)}{2 b d (a+b \tan (c+d x))^3}-\frac {\frac {a}{3 b d (a+b \tan (c+d x))^3}+\frac {2 b \left (-\frac {a \left (-\frac {i \log (i-\tan (c+d x))}{2 (a+i b)^4}+\frac {i \log (i+\tan (c+d x))}{2 (a-i b)^4}+\frac {4 a (a-b) b (a+b) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4}-\frac {b}{3 \left (a^2+b^2\right ) (a+b \tan (c+d x))^3}-\frac {a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}-\frac {b \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3 (a+b \tan (c+d x))}\right )}{b}+\frac {-\frac {\log (i-\tan (c+d x))}{2 (i a-b)^3}+\frac {\log (i+\tan (c+d x))}{2 (i a+b)^3}+\frac {b \left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}-\frac {b}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {2 a b}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}}{b}\right )}{d}}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^3/(a + b*Tan[c + d*x])^4,x]

[Out]

-1/2*Tan[c + d*x]/(b*d*(a + b*Tan[c + d*x])^3) - (a/(3*b*d*(a + b*Tan[c + d*x])^3) + (2*b*(-((a*(((-1/2*I)*Log
[I - Tan[c + d*x]])/(a + I*b)^4 + ((I/2)*Log[I + Tan[c + d*x]])/(a - I*b)^4 + (4*a*(a - b)*b*(a + b)*Log[a + b
*Tan[c + d*x]])/(a^2 + b^2)^4 - b/(3*(a^2 + b^2)*(a + b*Tan[c + d*x])^3) - (a*b)/((a^2 + b^2)^2*(a + b*Tan[c +
d*x])^2) - (b*(3*a^2 - b^2))/((a^2 + b^2)^3*(a + b*Tan[c + d*x]))))/b) + (-1/2*Log[I - Tan[c + d*x]]/(I*a - b
)^3 + Log[I + Tan[c + d*x]]/(2*(I*a + b)^3) + (b*(3*a^2 - b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^3 - b/(2*(
a^2 + b^2)*(a + b*Tan[c + d*x])^2) - (2*a*b)/((a^2 + b^2)^2*(a + b*Tan[c + d*x])))/b))/d)/(2*b)

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Maple [A]
time = 0.22, size = 206, normalized size = 1.09

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 a^{3} b +4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {a \left (a^{2}-3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a^{2} \left (a^{2}+3 b^{2}\right )}{2 \left (a^{2}+b^{2}\right )^{2} b^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{3}}{3 b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}}{d}\)	\(206\)
default	\(\frac {\frac {\frac {\left (-a^{4}+6 a^{2} b^{2}-b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-4 a^{3} b +4 a \,b^{3}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{4}}-\frac {a \left (a^{2}-3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}-\frac {a^{2} \left (a^{2}+3 b^{2}\right )}{2 \left (a^{2}+b^{2}\right )^{2} b^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{3}}{3 b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{3}}}{d}\)	\(206\)
norman	\(\frac {\frac {\left (a^{6}+6 a^{4} b^{2}-3 a^{2} b^{4}\right ) \left (\tan ^{2}\left (d x +c \right )\right )}{2 a d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {a \left (-3 a^{4} b +a^{2} b^{3}\right )}{3 d b \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {b \left (a^{6}+8 a^{4} b^{2}-9 a^{2} b^{4}\right ) \left (\tan ^{3}\left (d x +c \right )\right )}{6 a^{2} d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {4 \left (a^{2}-b^{2}\right ) a^{4} b x}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {12 b^{2} \left (a^{2}-b^{2}\right ) a^{3} x \tan \left (d x +c \right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {12 b^{3} \left (a^{2}-b^{2}\right ) a^{2} x \left (\tan ^{2}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}-\frac {4 b^{4} \left (a^{2}-b^{2}\right ) a x \left (\tan ^{3}\left (d x +c \right )\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{3}}+\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {\left (a^{4}-6 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\)	\(540\)
risch	\(-\frac {i x}{4 i a^{3} b -4 i a \,b^{3}-a^{4}+6 a^{2} b^{2}-b^{4}}-\frac {2 i a^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}+\frac {12 i a^{2} b^{2} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {2 i b^{4} x}{a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}}-\frac {2 i a^{4} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {12 i a^{2} b^{2} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {2 i b^{4} c}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {2 i a \left (-18 i a \,b^{4}+9 i a \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-18 i a^{3} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+9 a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}-6 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+9 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+9 i a \,b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-3 i a^{5} {\mathrm e}^{2 i \left (d x +c \right )}+6 i a^{3} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+24 a^{4} b \,{\mathrm e}^{2 i \left (d x +c \right )}+6 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-18 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+26 i a^{3} b^{2}-3 i a^{5} {\mathrm e}^{4 i \left (d x +c \right )}+13 a^{4} b -22 a^{2} b^{3}+9 b^{5}\right )}{3 \left (-i a +b \right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}+i a \,{\mathrm e}^{2 i \left (d x +c \right )}-b +i a \right )^{3} d \left (i a +b \right )^{4}}+\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}-\frac {6 a^{2} b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) b^{4}}{d \left (a^{8}+4 a^{6} b^{2}+6 a^{4} b^{4}+4 a^{2} b^{6}+b^{8}\right )}\)	\(790\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^3/(a+b*tan(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/(a^2+b^2)^4*(1/2*(-a^4+6*a^2*b^2-b^4)*ln(1+tan(d*x+c)^2)+(-4*a^3*b+4*a*b^3)*arctan(tan(d*x+c)))+(a^4-6*
a^2*b^2+b^4)/(a^2+b^2)^4*ln(a+b*tan(d*x+c))-a*(a^2-3*b^2)/(a^2+b^2)^3/(a+b*tan(d*x+c))-1/2*a^2*(a^2+3*b^2)/(a^
2+b^2)^2/b^2/(a+b*tan(d*x+c))^2+1/3*a^3/b^2/(a^2+b^2)/(a+b*tan(d*x+c))^3)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 402 vs. \(2 (185) = 370\).
time = 0.50, size = 402, normalized size = 2.13 \begin {gather*} -\frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {a^{7} + 14 \, a^{5} b^{2} - 11 \, a^{3} b^{4} + 6 \, {\left (a^{3} b^{4} - 3 \, a b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{6} b + 8 \, a^{4} b^{3} - 9 \, a^{2} b^{5}\right )} \tan \left (d x + c\right )}{a^{9} b^{2} + 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} + a^{3} b^{8} + {\left (a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{4} + 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} + a b^{10}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b^{3} + 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} + a^{2} b^{9}\right )} \tan \left (d x + c\right )}}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^3/(a+b*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/6*(24*(a^3*b - a*b^3)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 6*(a^4 - 6*a^2*b^2 + b^4)
*log(b*tan(d*x + c) + a)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 3*(a^4 - 6*a^2*b^2 + b^4)*log(tan(d
*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + (a^7 + 14*a^5*b^2 - 11*a^3*b^4 + 6*(a^3*b^4 -
3*a*b^6)*tan(d*x + c)^2 + 3*(a^6*b + 8*a^4*b^3 - 9*a^2*b^5)*tan(d*x + c))/(a^9*b^2 + 3*a^7*b^4 + 3*a^5*b^6 +
a^3*b^8 + (a^6*b^5 + 3*a^4*b^7 + 3*a^2*b^9 + b^11)*tan(d*x + c)^3 + 3*(a^7*b^4 + 3*a^5*b^6 + 3*a^3*b^8 + a*b^1
0)*tan(d*x + c)^2 + 3*(a^8*b^3 + 3*a^6*b^5 + 3*a^4*b^7 + a^2*b^9)*tan(d*x + c)))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (185) = 370\).
time = 1.06, size = 526, normalized size = 2.78 \begin {gather*} \frac {3 \, a^{7} - 30 \, a^{5} b^{2} + 11 \, a^{3} b^{4} + {\left (a^{6} b + 18 \, a^{4} b^{3} - 27 \, a^{2} b^{5} - 24 \, {\left (a^{3} b^{4} - a b^{6}\right )} d x\right )} \tan \left (d x + c\right )^{3} - 24 \, {\left (a^{6} b - a^{4} b^{3}\right )} d x + 3 \, {\left (a^{7} + 16 \, a^{5} b^{2} - 23 \, a^{3} b^{4} + 6 \, a b^{6} - 24 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{7} - 6 \, a^{5} b^{2} + a^{3} b^{4} + {\left (a^{4} b^{3} - 6 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{5} b^{2} - 6 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{6} b - 6 \, a^{4} b^{3} + a^{2} b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 3 \, {\left (9 \, a^{6} b - 26 \, a^{4} b^{3} + 9 \, a^{2} b^{5} - 24 \, {\left (a^{5} b^{2} - a^{3} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{6 \, {\left ({\left (a^{8} b^{3} + 4 \, a^{6} b^{5} + 6 \, a^{4} b^{7} + 4 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} b^{2} + 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} + 4 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right )^{2} + 3 \, {\left (a^{10} b + 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} + 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} + 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^3/(a+b*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/6*(3*a^7 - 30*a^5*b^2 + 11*a^3*b^4 + (a^6*b + 18*a^4*b^3 - 27*a^2*b^5 - 24*(a^3*b^4 - a*b^6)*d*x)*tan(d*x +
c)^3 - 24*(a^6*b - a^4*b^3)*d*x + 3*(a^7 + 16*a^5*b^2 - 23*a^3*b^4 + 6*a*b^6 - 24*(a^4*b^3 - a^2*b^5)*d*x)*tan
(d*x + c)^2 + 3*(a^7 - 6*a^5*b^2 + a^3*b^4 + (a^4*b^3 - 6*a^2*b^5 + b^7)*tan(d*x + c)^3 + 3*(a^5*b^2 - 6*a^3*b
^4 + a*b^6)*tan(d*x + c)^2 + 3*(a^6*b - 6*a^4*b^3 + a^2*b^5)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan
(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)) + 3*(9*a^6*b - 26*a^4*b^3 + 9*a^2*b^5 - 24*(a^5*b^2 - a^3*b^4)*d*x)*tan
(d*x + c))/((a^8*b^3 + 4*a^6*b^5 + 6*a^4*b^7 + 4*a^2*b^9 + b^11)*d*tan(d*x + c)^3 + 3*(a^9*b^2 + 4*a^7*b^4 + 6
*a^5*b^6 + 4*a^3*b^8 + a*b^10)*d*tan(d*x + c)^2 + 3*(a^10*b + 4*a^8*b^3 + 6*a^6*b^5 + 4*a^4*b^7 + a^2*b^9)*d*t
an(d*x + c) + (a^11 + 4*a^9*b^2 + 6*a^7*b^4 + 4*a^5*b^6 + a^3*b^8)*d)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**3/(a+b*tan(d*x+c))**4,x)

[Out]

Exception raised: AttributeError >> 'NoneType' object has no attribute 'primitive'

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (185) = 370\).
time = 1.10, size = 400, normalized size = 2.12 \begin {gather*} -\frac {\frac {24 \, {\left (a^{3} b - a b^{3}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {6 \, {\left (a^{4} b - 6 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} + \frac {11 \, a^{4} b^{5} \tan \left (d x + c\right )^{3} - 66 \, a^{2} b^{7} \tan \left (d x + c\right )^{3} + 11 \, b^{9} \tan \left (d x + c\right )^{3} + 39 \, a^{5} b^{4} \tan \left (d x + c\right )^{2} - 210 \, a^{3} b^{6} \tan \left (d x + c\right )^{2} + 15 \, a b^{8} \tan \left (d x + c\right )^{2} + 3 \, a^{8} b \tan \left (d x + c\right ) + 60 \, a^{6} b^{3} \tan \left (d x + c\right ) - 201 \, a^{4} b^{5} \tan \left (d x + c\right ) + 6 \, a^{2} b^{7} \tan \left (d x + c\right ) + a^{9} + 26 \, a^{7} b^{2} - 63 \, a^{5} b^{4}}{{\left (a^{8} b^{2} + 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}}}{6 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^3/(a+b*tan(d*x+c))^4,x, algorithm="giac")

[Out]

-1/6*(24*(a^3*b - a*b^3)*(d*x + c)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) + 3*(a^4 - 6*a^2*b^2 + b^4)
*log(tan(d*x + c)^2 + 1)/(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8) - 6*(a^4*b - 6*a^2*b^3 + b^5)*log(abs
(b*tan(d*x + c) + a))/(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9) + (11*a^4*b^5*tan(d*x + c)^3 - 66*a^2*
b^7*tan(d*x + c)^3 + 11*b^9*tan(d*x + c)^3 + 39*a^5*b^4*tan(d*x + c)^2 - 210*a^3*b^6*tan(d*x + c)^2 + 15*a*b^8
*tan(d*x + c)^2 + 3*a^8*b*tan(d*x + c) + 60*a^6*b^3*tan(d*x + c) - 201*a^4*b^5*tan(d*x + c) + 6*a^2*b^7*tan(d*
x + c) + a^9 + 26*a^7*b^2 - 63*a^5*b^4)/((a^8*b^2 + 4*a^6*b^4 + 6*a^4*b^6 + 4*a^2*b^8 + b^10)*(b*tan(d*x + c)
+ a)^3))/d

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Mupad [B]
time = 4.64, size = 359, normalized size = 1.90 \begin {gather*} \frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {1}{{\left (a^2+b^2\right )}^2}-\frac {8\,b^2}{{\left (a^2+b^2\right )}^3}+\frac {8\,b^4}{{\left (a^2+b^2\right )}^4}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{2\,d\,\left (a^4+a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2-a\,b^3\,4{}\mathrm {i}+b^4\right )}-\frac {\frac {a\,\left (a^6+14\,a^4\,b^2-11\,a^2\,b^4\right )}{6\,b^2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (3\,a\,b^4-a^3\,b^2\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (a^6+8\,a^4\,b^2-9\,a^2\,b^4\right )}{2\,b\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (a^3+3\,a^2\,b\,\mathrm {tan}\left (c+d\,x\right )+3\,a\,b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^3\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d\,\left (a^4\,1{}\mathrm {i}+4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}-4\,a\,b^3+b^4\,1{}\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^3/(a + b*tan(c + d*x))^4,x)

[Out]

(log(a + b*tan(c + d*x))*(1/(a^2 + b^2)^2 - (8*b^2)/(a^2 + b^2)^3 + (8*b^4)/(a^2 + b^2)^4))/d - (log(tan(c + d
*x) + 1i)*1i)/(2*d*(4*a^3*b - 4*a*b^3 + a^4*1i + b^4*1i - a^2*b^2*6i)) - log(tan(c + d*x) - 1i)/(2*d*(a^3*b*4i
- a*b^3*4i + a^4 + b^4 - 6*a^2*b^2)) - ((a*(a^6 - 11*a^2*b^4 + 14*a^4*b^2))/(6*b^2*(a^6 + b^6 + 3*a^2*b^4 + 3
*a^4*b^2)) - (tan(c + d*x)^2*(3*a*b^4 - a^3*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (tan(c + d*x)*(a^6 - 9
*a^2*b^4 + 8*a^4*b^2))/(2*b*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(a^3 + b^3*tan(c + d*x)^3 + 3*a*b^2*tan(c
+ d*x)^2 + 3*a^2*b*tan(c + d*x)))

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