Optimal. Leaf size=217 \[ -\frac{a^4 b}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left (\left (a^2-b^2\right ) \tan (c+d x)+2 a b\right )}{4 d \left (a^2+b^2\right )^2}-\frac{\cos ^2(c+d x) \left (\left (-12 a^2 b^2+5 a^4-b^4\right ) \tan (c+d x)+16 a^3 b\right )}{8 d \left (a^2+b^2\right )^3}+\frac{2 a^3 b \left (a^2-2 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{x \left (-33 a^4 b^2+13 a^2 b^4+3 a^6+b^6\right )}{8 \left (a^2+b^2\right )^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.561945, antiderivative size = 217, 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} \[ -\frac{a^4 b}{d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left (\left (a^2-b^2\right ) \tan (c+d x)+2 a b\right )}{4 d \left (a^2+b^2\right )^2}-\frac{\cos ^2(c+d x) \left (\left (-12 a^2 b^2+5 a^4-b^4\right ) \tan (c+d x)+16 a^3 b\right )}{8 d \left (a^2+b^2\right )^3}+\frac{2 a^3 b \left (a^2-2 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac{x \left (-33 a^4 b^2+13 a^2 b^4+3 a^6+b^6\right )}{8 \left (a^2+b^2\right )^4} \]
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))^2} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{x^4}{(a+x)^2 \left (b^2+x^2\right )^3} \, dx,x,b \tan (c+d x)\right )}{d}\\ &=\frac{\cos ^4(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac{\operatorname{Subst}\left (\int \frac{\frac{a^2 b^4 \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2}-\frac{2 a b^4 \left (3 a^2+b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac{b^2 \left (4 a^4+11 a^2 b^2+b^4\right ) x^2}{\left (a^2+b^2\right )^2}}{(a+x)^2 \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 (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac{\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}+\frac{\operatorname{Subst}\left (\int \frac{\frac{a^2 b^4 \left (3 a^4-12 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^3}-\frac{2 a b^4 \left (5 a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}-\frac{b^4 \left (5 a^4-12 a^2 b^2-b^4\right ) x^2}{\left (a^2+b^2\right )^3}}{(a+x)^2 \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 (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac{\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}+\frac{\operatorname{Subst}\left (\int \left (\frac{8 a^4 b^4}{\left (a^2+b^2\right )^3 (a+x)^2}+\frac{16 a^3 b^4 \left (a^2-2 b^2\right )}{\left (a^2+b^2\right )^4 (a+x)}+\frac{b^4 \left (3 a^6-33 a^4 b^2+13 a^2 b^4+b^6-16 a^3 \left (a^2-2 b^2\right ) x\right )}{\left (a^2+b^2\right )^4 \left (b^2+x^2\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{8 b^3 d}\\ &=\frac{2 a^3 b \left (a^2-2 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac{a^4 b}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac{\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}+\frac{b \operatorname{Subst}\left (\int \frac{3 a^6-33 a^4 b^2+13 a^2 b^4+b^6-16 a^3 \left (a^2-2 b^2\right ) x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^4 d}\\ &=\frac{2 a^3 b \left (a^2-2 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac{a^4 b}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac{\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}-\frac{\left (2 a^3 b \left (a^2-2 b^2\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 )^4 d}+\frac{\left (b \left (3 a^6-33 a^4 b^2+13 a^2 b^4+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 )^4 d}\\ &=\frac{\left (3 a^6-33 a^4 b^2+13 a^2 b^4+b^6\right ) x}{8 \left (a^2+b^2\right )^4}+\frac{2 a^3 b \left (a^2-2 b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac{2 a^3 b \left (a^2-2 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}-\frac{a^4 b}{\left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}+\frac{\cos ^4(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^2 d}-\frac{\cos ^2(c+d x) \left (16 a^3 b+\left (5 a^4-12 a^2 b^2-b^4\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}\\ \end{align*}
Mathematica [A] time = 3.80693, size = 373, normalized size = 1.72 \[ \frac{b \left (\frac{2 \left (a^2+b^2\right ) \left (3 a^2 b^2-2 a^4+b^4\right ) \sin (2 (c+d x))}{b}-16 a^3 \left (a^2+b^2\right ) \cos ^2(c+d x)+4 a \left (a^2+b^2\right )^2 \cos ^4(c+d x)+\frac{4 \left (a^2+b^2\right ) \left (3 a^2 b^2-2 a^4+b^4\right ) \tan ^{-1}(\tan (c+d x))}{b}-\frac{8 a^4 \left (a^2+b^2\right )}{a+b \tan (c+d x)}-4 a^3 \left (\frac{5 a b^2-a^3}{\sqrt{-b^2}}+2 a^2-4 b^2\right ) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )+16 a^3 \left (a^2-2 b^2\right ) \log (a+b \tan (c+d x))-4 a^3 \left (\frac{a^3-5 a b^2}{\sqrt{-b^2}}+2 a^2-4 b^2\right ) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )+\frac{3 \left (a^2-b^2\right ) \left (a^2+b^2\right )^2 \left (\sin (2 (c+d x))+2 \tan ^{-1}(\tan (c+d x))\right )}{2 b}+\frac{2 \left (a^2-b^2\right ) \left (a^2+b^2\right )^2 \sin (c+d x) \cos ^3(c+d x)}{b}\right )}{8 d \left (a^2+b^2\right )^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.099, size = 724, normalized size = 3.3 \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 = 1.53015, size = 684, normalized size = 3.15 \begin{align*} \frac{\frac{{\left (3 \, a^{6} - 33 \, a^{4} b^{2} + 13 \, a^{2} b^{4} + b^{6}\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{16 \,{\left (a^{5} b - 2 \, a^{3} b^{3}\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{8 \,{\left (a^{5} b - 2 \, a^{3} b^{3}\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{20 \, a^{4} b - 4 \, a^{2} b^{3} +{\left (13 \, a^{4} b - 12 \, a^{2} b^{3} - b^{5}\right )} \tan \left (d x + c\right )^{4} +{\left (5 \, a^{5} + 4 \, a^{3} b^{2} - a b^{4}\right )} \tan \left (d x + c\right )^{3} +{\left (35 \, a^{4} b - 12 \, a^{2} b^{3} + b^{5}\right )} \tan \left (d x + c\right )^{2} + 3 \,{\left (a^{5} - a b^{4}\right )} \tan \left (d x + c\right )}{a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6} +{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{5} +{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{4} + 2 \,{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{3} + 2 \,{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )^{2} +{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.63226, size = 986, normalized size = 4.54 \begin{align*} \frac{4 \,{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{5} - 6 \,{\left (3 \, a^{6} b + 7 \, a^{4} b^{3} + 5 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{3} +{\left (3 \, a^{6} b + 8 \, a^{4} b^{3} + 23 \, a^{2} b^{5} + 2 \, b^{7} + 2 \,{\left (3 \, a^{7} - 33 \, a^{5} b^{2} + 13 \, a^{3} b^{4} + a b^{6}\right )} d x\right )} \cos \left (d x + c\right ) + 16 \,{\left ({\left (a^{6} b - 2 \, a^{4} b^{3}\right )} \cos \left (d x + c\right ) +{\left (a^{5} b^{2} - 2 \, a^{3} b^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) +{\left (29 \, a^{5} b^{2} + 10 \, a^{3} b^{4} - 3 \, a b^{6} + 4 \,{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (3 \, a^{6} b - 33 \, a^{4} b^{3} + 13 \, a^{2} b^{5} + b^{7}\right )} d x - 2 \,{\left (5 \, a^{7} + 9 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \,{\left ({\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} d \cos \left (d x + c\right ) +{\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} d \sin \left (d x + c\right )\right )}} \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 = 1.21571, size = 693, normalized size = 3.19 \begin{align*} \frac{\frac{{\left (3 \, a^{6} - 33 \, a^{4} b^{2} + 13 \, a^{2} b^{4} + b^{6}\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{8 \,{\left (a^{5} b - 2 \, a^{3} b^{3}\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{16 \,{\left (a^{5} b^{2} - 2 \, a^{3} b^{4}\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{8 \,{\left (2 \, a^{5} b^{2} \tan \left (d x + c\right ) - 4 \, a^{3} b^{4} \tan \left (d x + c\right ) + 3 \, a^{6} b - 3 \, a^{4} b^{3}\right )}}{{\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 ) + a\right )}} + \frac{12 \, a^{5} b \tan \left (d x + c\right )^{4} - 24 \, a^{3} b^{3} \tan \left (d x + c\right )^{4} - 5 \, a^{6} \tan \left (d x + c\right )^{3} + 7 \, a^{4} b^{2} \tan \left (d x + c\right )^{3} + 13 \, a^{2} b^{4} \tan \left (d x + c\right )^{3} + b^{6} \tan \left (d x + c\right )^{3} + 8 \, a^{5} b \tan \left (d x + c\right )^{2} - 64 \, a^{3} b^{3} \tan \left (d x + c\right )^{2} - 3 \, a^{6} \tan \left (d x + c\right ) + 9 \, a^{4} b^{2} \tan \left (d x + c\right ) + 11 \, a^{2} b^{4} \tan \left (d x + c\right ) - b^{6} \tan \left (d x + c\right ) - 32 \, a^{3} b^{3} + 4 \, a b^{5}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )}{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]