Optimal. Leaf size=94 \[ \frac {3 a \left (a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {3 a \sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{8 d}+\frac {\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^3}{4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2668, 729, 723, 206} \[ \frac {3 a \left (a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {3 a \sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{8 d}+\frac {\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^3}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 723
Rule 729
Rule 2668
Rubi steps
\begin {align*} \int \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {b^5 \operatorname {Subst}\left (\int \frac {(a+x)^3}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^3 \tan (c+d x)}{4 d}+\frac {\left (3 a b^3\right ) \operatorname {Subst}\left (\int \frac {(a+x)^2}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d}\\ &=\frac {3 a \sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))}{8 d}+\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^3 \tan (c+d x)}{4 d}+\frac {\left (3 a b \left (a^2-b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac {3 a \left (a^2-b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {3 a \sec ^2(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))}{8 d}+\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^3 \tan (c+d x)}{4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 4.04, size = 318, normalized size = 3.38 \[ \frac {-6 a \left (a^2-b^2\right )^3 (\log (1-\sin (c+d x))-\log (\sin (c+d x)+1))+16 a^4 b \left (3 a^2-2 b^2\right ) \tan ^2(c+d x)+16 a^2 b \sec ^2(c+d x) \left (\left (2 a^4-5 a^2 b^2+3 b^4\right ) \tan ^2(c+d x)-a^4\right )+8 b^3 \left (4 a^4-5 a^2 b^2+b^4\right ) \tan ^4(c+d x)+4 a \tan (c+d x) \sec (c+d x) \left (3 \left (a^6-5 a^4 b^2\right )+4 b^2 \left (3 a^4-5 a^2 b^2+2 b^4\right ) \tan ^2(c+d x)\right )+a \left (8 a^6-22 a^4 b^2+29 a^2 b^4-3 b^6\right ) \tan (c+d x) \sec ^3(c+d x)+a b \sec ^4(c+d x) \left (-8 a^5+8 a^3 b^2+\left (18 a^4 b-11 a^2 b^3+5 b^5\right ) \sin (3 (c+d x))\right )}{32 d \left (a^2-b^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 138, normalized size = 1.47 \[ \frac {3 \, {\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 8 \, b^{3} \cos \left (d x + c\right )^{2} + 12 \, a^{2} b + 4 \, b^{3} + 2 \, {\left (2 \, a^{3} + 6 \, a b^{2} + 3 \, {\left (a^{3} - a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.79, size = 139, normalized size = 1.48 \[ \frac {3 \, {\left (a^{3} - a b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 3 \, {\left (a^{3} - a b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (3 \, a^{3} \sin \left (d x + c\right )^{3} - 3 \, a b^{2} \sin \left (d x + c\right )^{3} - 4 \, b^{3} \sin \left (d x + c\right )^{2} - 5 \, a^{3} \sin \left (d x + c\right ) - 3 \, a b^{2} \sin \left (d x + c\right ) - 6 \, a^{2} b + 2 \, b^{3}\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.27, size = 195, normalized size = 2.07 \[ \frac {a^{3} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 a^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {3 a^{2} b}{4 d \cos \left (d x +c \right )^{4}}+\frac {3 a \,b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}+\frac {3 a \,b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}+\frac {3 a \,b^{2} \sin \left (d x +c \right )}{8 d}-\frac {3 a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 136, normalized size = 1.45 \[ \frac {3 \, {\left (a^{3} - a b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (a^{3} - a b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + \frac {2 \, {\left (4 \, b^{3} \sin \left (d x + c\right )^{2} - 3 \, {\left (a^{3} - a b^{2}\right )} \sin \left (d x + c\right )^{3} + 6 \, a^{2} b - 2 \, b^{3} + {\left (5 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.17, size = 114, normalized size = 1.21 \[ \frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {3\,a\,b^2}{8}-\frac {3\,a^3}{8}\right )+\frac {3\,a^2\,b}{4}-\frac {b^3}{4}+\sin \left (c+d\,x\right )\,\left (\frac {5\,a^3}{8}+\frac {3\,a\,b^2}{8}\right )+\frac {b^3\,{\sin \left (c+d\,x\right )}^2}{2}}{d\,\left ({\sin \left (c+d\,x\right )}^4-2\,{\sin \left (c+d\,x\right )}^2+1\right )}+\frac {3\,a\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )\,\left (a^2-b^2\right )}{8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________