Optimal. Leaf size=285 \[ -\frac{a^4 b}{2 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^2}-\frac{2 a^3 b \left (a^2-2 b^2\right )}{d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))}-\frac{a \cos ^2(c+d x) \left (\left (-34 a^2 b^2+5 a^4+9 b^4\right ) \tan (c+d x)+24 a b \left (a^2-b^2\right )\right )}{8 d \left (a^2+b^2\right )^4}+\frac{\cos ^4(c+d x) \left (a \left (a^2-3 b^2\right ) \tan (c+d x)+b \left (3 a^2-b^2\right )\right )}{4 d \left (a^2+b^2\right )^3}+\frac{3 a^2 b \left (-5 a^2 b^2+a^4+2 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^5}+\frac{3 a x \left (-25 a^4 b^2+35 a^2 b^4+a^6-3 b^6\right )}{8 \left (a^2+b^2\right )^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.849748, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3516, 1647, 1629, 635, 203, 260} \[ -\frac{a^4 b}{2 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^2}-\frac{2 a^3 b \left (a^2-2 b^2\right )}{d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))}-\frac{a \cos ^2(c+d x) \left (\left (-34 a^2 b^2+5 a^4+9 b^4\right ) \tan (c+d x)+24 a b \left (a^2-b^2\right )\right )}{8 d \left (a^2+b^2\right )^4}+\frac{\cos ^4(c+d x) \left (a \left (a^2-3 b^2\right ) \tan (c+d x)+b \left (3 a^2-b^2\right )\right )}{4 d \left (a^2+b^2\right )^3}+\frac{3 a^2 b \left (-5 a^2 b^2+a^4+2 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^5}+\frac{3 a x \left (-25 a^4 b^2+35 a^2 b^4+a^6-3 b^6\right )}{8 \left (a^2+b^2\right )^5} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3516
Rule 1647
Rule 1629
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{\sin ^4(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{x^4}{(a+x)^3 \left (b^2+x^2\right )^3} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{\cos ^4(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 d}-\frac{\operatorname{Subst}\left (\int \frac{\frac{a^4 b^4 \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3}-\frac{a^3 b^4 \left (9 a^2+5 b^2\right ) x}{\left (a^2+b^2\right )^3}-\frac{a^2 b^2 \left (4 a^4+21 a^2 b^2-3 b^4\right ) x^2}{\left (a^2+b^2\right )^3}-\frac{3 a b^4 \left (a^2-3 b^2\right ) x^3}{\left (a^2+b^2\right )^3}}{(a+x)^3 \left (b^2+x^2\right )^2} \, dx,x,b \tan (c+d x)\right )}{4 b d}\\ &=\frac{\cos ^4(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 d}-\frac{a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}+\frac{\operatorname{Subst}\left (\int \frac{\frac{3 a^4 b^4 \left (a^4-10 a^2 b^2+5 b^4\right )}{\left (a^2+b^2\right )^4}-\frac{a^3 b^4 \left (15 a^4+26 a^2 b^2-37 b^4\right ) x}{\left (a^2+b^2\right )^4}-\frac{3 a^2 b^4 \left (5 a^4-18 a^2 b^2-7 b^4\right ) x^2}{\left (a^2+b^2\right )^4}-\frac{a b^4 \left (5 a^4-34 a^2 b^2+9 b^4\right ) x^3}{\left (a^2+b^2\right )^4}}{(a+x)^3 \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{8 b^3 d}\\ &=\frac{\cos ^4(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 d}-\frac{a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}+\frac{\operatorname{Subst}\left (\int \left (\frac{8 a^4 b^4}{\left (a^2+b^2\right )^3 (a+x)^3}+\frac{16 a^3 b^4 \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 (a+x)^2}+\frac{24 a^2 b^4 \left (a^4-5 a^2 b^2+2 b^4\right )}{\left (a^2+b^2\right )^5 (a+x)}+\frac{3 a b^4 \left (a^6-25 a^4 b^2+35 a^2 b^4-3 b^6-8 a \left (a^4-5 a^2 b^2+2 b^4\right ) x\right )}{\left (a^2+b^2\right )^5 \left (b^2+x^2\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{8 b^3 d}\\ &=\frac{3 a^2 b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5 d}-\frac{a^4 b}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac{2 a^3 b \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 d}-\frac{a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}+\frac{(3 a b) \operatorname{Subst}\left (\int \frac{a^6-25 a^4 b^2+35 a^2 b^4-3 b^6-8 a \left (a^4-5 a^2 b^2+2 b^4\right ) x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^5 d}\\ &=\frac{3 a^2 b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5 d}-\frac{a^4 b}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac{2 a^3 b \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 d}-\frac{a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}-\frac{\left (3 a^2 b \left (a^4-5 a^2 b^2+2 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{\left (a^2+b^2\right )^5 d}+\frac{\left (3 a b \left (a^6-25 a^4 b^2+35 a^2 b^4-3 b^6\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^5 d}\\ &=\frac{3 a \left (a^6-25 a^4 b^2+35 a^2 b^4-3 b^6\right ) x}{8 \left (a^2+b^2\right )^5}+\frac{3 a^2 b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^5 d}+\frac{3 a^2 b \left (a^4-5 a^2 b^2+2 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5 d}-\frac{a^4 b}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))^2}-\frac{2 a^3 b \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 d}-\frac{a \cos ^2(c+d x) \left (24 a b \left (a^2-b^2\right )+\left (5 a^4-34 a^2 b^2+9 b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}\\ \end{align*}
Mathematica [A] time = 6.43835, size = 501, normalized size = 1.76 \[ \frac{b \left (-\frac{3 a^2 (a-b) (a+b) \cos ^2(c+d x)}{\left (a^2+b^2\right )^4}+\frac{\left (3 a^2-b^2\right ) \cos ^4(c+d x)}{4 \left (a^2+b^2\right )^3}-\frac{a^3 \left (a^2-5 b^2\right ) \tan ^{-1}(\tan (c+d x))}{b \left (a^2+b^2\right )^4}-\frac{a^4}{2 \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^2}-\frac{2 a^3 \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 (a+b \tan (c+d x))}-\frac{a^2 \left (-15 a^2 b^2-\frac{-13 a^3 b^2+a^5+10 a b^4}{\sqrt{-b^2}}+3 a^4+6 b^4\right ) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5}+\frac{3 a^2 \left (-5 a^2 b^2+a^4+2 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5}-\frac{a^2 \left (-15 a^2 b^2+\frac{-13 a^3 b^2+a^5+10 a b^4}{\sqrt{-b^2}}+3 a^4+6 b^4\right ) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5}+\frac{a \left (a^2-3 b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b \left (a^2+b^2\right )^3}-\frac{a^3 \left (a^2-5 b^2\right ) \sin (c+d x) \cos (c+d x)}{b \left (a^2+b^2\right )^4}+\frac{3 a \left (a^2-3 b^2\right ) \left (\frac{\tan ^{-1}(\tan (c+d x))}{b}+\frac{\sin (c+d x) \cos (c+d x)}{b}\right )}{8 \left (a^2+b^2\right )^3}\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.109, size = 882, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.81091, size = 1004, normalized size = 3.52 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 3.09478, size = 1569, normalized size = 5.51 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.27008, size = 794, normalized size = 2.79 \begin{align*} \frac{\frac{3 \,{\left (a^{7} - 25 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 3 \, a b^{6}\right )}{\left (d x + c\right )}}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} - \frac{12 \,{\left (a^{6} b - 5 \, a^{4} b^{3} + 2 \, a^{2} b^{5}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} + \frac{24 \,{\left (a^{6} b^{2} - 5 \, a^{4} b^{4} + 2 \, a^{2} b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{10} b + 5 \, a^{8} b^{3} + 10 \, a^{6} b^{5} + 10 \, a^{4} b^{7} + 5 \, a^{2} b^{9} + b^{11}} - \frac{21 \, a^{5} b^{2} \tan \left (d x + c\right )^{5} - 66 \, a^{3} b^{4} \tan \left (d x + c\right )^{5} + 9 \, a b^{6} \tan \left (d x + c\right )^{5} + 30 \, a^{6} b \tan \left (d x + c\right )^{4} - 72 \, a^{4} b^{3} \tan \left (d x + c\right )^{4} - 6 \, a^{2} b^{5} \tan \left (d x + c\right )^{4} + 5 \, a^{7} \tan \left (d x + c\right )^{3} + 49 \, a^{5} b^{2} \tan \left (d x + c\right )^{3} - 133 \, a^{3} b^{4} \tan \left (d x + c\right )^{3} + 15 \, a b^{6} \tan \left (d x + c\right )^{3} + 70 \, a^{6} b \tan \left (d x + c\right )^{2} - 122 \, a^{4} b^{3} \tan \left (d x + c\right )^{2} + 2 \, a^{2} b^{5} \tan \left (d x + c\right )^{2} + 2 \, b^{7} \tan \left (d x + c\right )^{2} + 3 \, a^{7} \tan \left (d x + c\right ) + 22 \, a^{5} b^{2} \tan \left (d x + c\right ) - 73 \, a^{3} b^{4} \tan \left (d x + c\right ) + 4 \, a b^{6} \tan \left (d x + c\right ) + 38 \, a^{6} b - 56 \, a^{4} b^{3} + 2 \, a^{2} b^{5}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}{\left (b \tan \left (d x + c\right )^{3} + a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right ) + a\right )}^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]