Optimal. Leaf size=155 \[ \frac{a \cos ^2(c+d x)}{2 d}+\frac{a \sec ^4(c+d x)}{4 d}-\frac{3 a \sec ^2(c+d x)}{2 d}-\frac{3 a \log (\cos (c+d x))}{d}-\frac{35 b \sin ^3(c+d x)}{24 d}-\frac{35 b \sin (c+d x)}{8 d}+\frac{b \sin ^3(c+d x) \tan ^4(c+d x)}{4 d}-\frac{7 b \sin ^3(c+d x) \tan ^2(c+d x)}{8 d}+\frac{35 b \tanh ^{-1}(\sin (c+d x))}{8 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.168642, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2834, 2590, 266, 43, 2592, 288, 302, 206} \[ \frac{a \cos ^2(c+d x)}{2 d}+\frac{a \sec ^4(c+d x)}{4 d}-\frac{3 a \sec ^2(c+d x)}{2 d}-\frac{3 a \log (\cos (c+d x))}{d}-\frac{35 b \sin ^3(c+d x)}{24 d}-\frac{35 b \sin (c+d x)}{8 d}+\frac{b \sin ^3(c+d x) \tan ^4(c+d x)}{4 d}-\frac{7 b \sin ^3(c+d x) \tan ^2(c+d x)}{8 d}+\frac{35 b \tanh ^{-1}(\sin (c+d x))}{8 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2834
Rule 2590
Rule 266
Rule 43
Rule 2592
Rule 288
Rule 302
Rule 206
Rubi steps
\begin{align*} \int \sin ^2(c+d x) (a+b \sin (c+d x)) \tan ^5(c+d x) \, dx &=a \int \sin ^2(c+d x) \tan ^5(c+d x) \, dx+b \int \sin ^3(c+d x) \tan ^5(c+d x) \, dx\\ &=-\frac{a \operatorname{Subst}\left (\int \frac{\left (1-x^2\right )^3}{x^5} \, dx,x,\cos (c+d x)\right )}{d}+\frac{b \operatorname{Subst}\left (\int \frac{x^8}{\left (1-x^2\right )^3} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{b \sin ^3(c+d x) \tan ^4(c+d x)}{4 d}-\frac{a \operatorname{Subst}\left (\int \frac{(1-x)^3}{x^3} \, dx,x,\cos ^2(c+d x)\right )}{2 d}-\frac{(7 b) \operatorname{Subst}\left (\int \frac{x^6}{\left (1-x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{4 d}\\ &=-\frac{7 b \sin ^3(c+d x) \tan ^2(c+d x)}{8 d}+\frac{b \sin ^3(c+d x) \tan ^4(c+d x)}{4 d}-\frac{a \operatorname{Subst}\left (\int \left (-1+\frac{1}{x^3}-\frac{3}{x^2}+\frac{3}{x}\right ) \, dx,x,\cos ^2(c+d x)\right )}{2 d}+\frac{(35 b) \operatorname{Subst}\left (\int \frac{x^4}{1-x^2} \, dx,x,\sin (c+d x)\right )}{8 d}\\ &=\frac{a \cos ^2(c+d x)}{2 d}-\frac{3 a \log (\cos (c+d x))}{d}-\frac{3 a \sec ^2(c+d x)}{2 d}+\frac{a \sec ^4(c+d x)}{4 d}-\frac{7 b \sin ^3(c+d x) \tan ^2(c+d x)}{8 d}+\frac{b \sin ^3(c+d x) \tan ^4(c+d x)}{4 d}+\frac{(35 b) \operatorname{Subst}\left (\int \left (-1-x^2+\frac{1}{1-x^2}\right ) \, dx,x,\sin (c+d x)\right )}{8 d}\\ &=\frac{a \cos ^2(c+d x)}{2 d}-\frac{3 a \log (\cos (c+d x))}{d}-\frac{3 a \sec ^2(c+d x)}{2 d}+\frac{a \sec ^4(c+d x)}{4 d}-\frac{35 b \sin (c+d x)}{8 d}-\frac{35 b \sin ^3(c+d x)}{24 d}-\frac{7 b \sin ^3(c+d x) \tan ^2(c+d x)}{8 d}+\frac{b \sin ^3(c+d x) \tan ^4(c+d x)}{4 d}+\frac{(35 b) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{8 d}\\ &=\frac{35 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \cos ^2(c+d x)}{2 d}-\frac{3 a \log (\cos (c+d x))}{d}-\frac{3 a \sec ^2(c+d x)}{2 d}+\frac{a \sec ^4(c+d x)}{4 d}-\frac{35 b \sin (c+d x)}{8 d}-\frac{35 b \sin ^3(c+d x)}{24 d}-\frac{7 b \sin ^3(c+d x) \tan ^2(c+d x)}{8 d}+\frac{b \sin ^3(c+d x) \tan ^4(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.588935, size = 156, normalized size = 1.01 \[ -\frac{a \left (2 \sin ^2(c+d x)-\sec ^4(c+d x)+6 \sec ^2(c+d x)+12 \log (\cos (c+d x))\right )}{4 d}-\frac{b \sin ^3(c+d x) \tan ^4(c+d x)}{3 d}-\frac{7 b \left (8 \sin (c+d x) \tan ^4(c+d x)+5 \left (6 \tan (c+d x) \sec ^3(c+d x)-8 \tan ^3(c+d x) \sec (c+d x)-3 \left (\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right )\right )\right )}{24 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.05, size = 219, normalized size = 1.4 \begin{align*}{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{8}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{6}}{2\,d}}-{\frac{3\,a \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}a}{2\,d}}-3\,{\frac{a\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{b \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{5\,b \left ( \sin \left ( dx+c \right ) \right ) ^{9}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{5\,b \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{8\,d}}-{\frac{7\,b \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{8\,d}}-{\frac{35\,b \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{24\,d}}-{\frac{35\,b\sin \left ( dx+c \right ) }{8\,d}}+{\frac{35\,b\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.981233, size = 178, normalized size = 1.15 \begin{align*} -\frac{16 \, b \sin \left (d x + c\right )^{3} + 24 \, a \sin \left (d x + c\right )^{2} + 3 \,{\left (24 \, a - 35 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \,{\left (24 \, a + 35 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + 144 \, b \sin \left (d x + c\right ) - \frac{6 \,{\left (13 \, b \sin \left (d x + c\right )^{3} + 12 \, a \sin \left (d x + c\right )^{2} - 11 \, b \sin \left (d x + c\right ) - 10 \, a\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.10069, size = 401, normalized size = 2.59 \begin{align*} \frac{24 \, a \cos \left (d x + c\right )^{6} - 3 \,{\left (24 \, a - 35 \, b\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (24 \, a + 35 \, b\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 12 \, a \cos \left (d x + c\right )^{4} - 72 \, a \cos \left (d x + c\right )^{2} + 2 \,{\left (8 \, b \cos \left (d x + c\right )^{6} - 80 \, b \cos \left (d x + c\right )^{4} - 39 \, b \cos \left (d x + c\right )^{2} + 6 \, b\right )} \sin \left (d x + c\right ) + 12 \, a}{48 \, d \cos \left (d x + c\right )^{4}} \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 [A] time = 1.26742, size = 182, normalized size = 1.17 \begin{align*} -\frac{16 \, b \sin \left (d x + c\right )^{3} + 24 \, a \sin \left (d x + c\right )^{2} + 3 \,{\left (24 \, a - 35 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 3 \,{\left (24 \, a + 35 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + 144 \, b \sin \left (d x + c\right ) - \frac{6 \,{\left (18 \, a \sin \left (d x + c\right )^{4} + 13 \, b \sin \left (d x + c\right )^{3} - 24 \, a \sin \left (d x + c\right )^{2} - 11 \, b \sin \left (d x + c\right ) + 8 \, a\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]