Optimal. Leaf size=171 \[ -\frac {a^3 \csc (c+d x)}{d}+\frac {b \sec ^4(c+d x) \left (a b \left (\frac {a^2}{b^2}+3\right ) \sin (c+d x)+3 a^2+b^2\right )}{4 d}+\frac {a b \sec ^2(c+d x) \left (b \left (\frac {7 a^2}{b^2}+9\right ) \sin (c+d x)+12 a\right )}{8 d}+\frac {3 a^2 b \log (\sin (c+d x))}{d}-\frac {3 a (a+b) (5 a+3 b) \log (1-\sin (c+d x))}{16 d}+\frac {3 a (5 a-3 b) (a-b) \log (\sin (c+d x)+1)}{16 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.37, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2837, 12, 1805, 1802} \[ \frac {b \sec ^4(c+d x) \left (a b \left (\frac {a^2}{b^2}+3\right ) \sin (c+d x)+3 a^2+b^2\right )}{4 d}+\frac {a b \sec ^2(c+d x) \left (b \left (\frac {7 a^2}{b^2}+9\right ) \sin (c+d x)+12 a\right )}{8 d}+\frac {3 a^2 b \log (\sin (c+d x))}{d}-\frac {a^3 \csc (c+d x)}{d}-\frac {3 a (a+b) (5 a+3 b) \log (1-\sin (c+d x))}{16 d}+\frac {3 a (5 a-3 b) (a-b) \log (\sin (c+d x)+1)}{16 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 1802
Rule 1805
Rule 2837
Rubi steps
\begin {align*} \int \csc ^2(c+d x) \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {b^5 \operatorname {Subst}\left (\int \frac {b^2 (a+x)^3}{x^2 \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b^7 \operatorname {Subst}\left (\int \frac {(a+x)^3}{x^2 \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b \sec ^4(c+d x) \left (3 a^2+b^2+a \left (3+\frac {a^2}{b^2}\right ) b \sin (c+d x)\right )}{4 d}-\frac {b^5 \operatorname {Subst}\left (\int \frac {-4 a^3-12 a^2 x-3 a \left (3+\frac {a^2}{b^2}\right ) x^2}{x^2 \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d}\\ &=\frac {b \sec ^4(c+d x) \left (3 a^2+b^2+a \left (3+\frac {a^2}{b^2}\right ) b \sin (c+d x)\right )}{4 d}+\frac {a b \sec ^2(c+d x) \left (12 a+\left (9+\frac {7 a^2}{b^2}\right ) b \sin (c+d x)\right )}{8 d}+\frac {b^3 \operatorname {Subst}\left (\int \frac {8 a^3+24 a^2 x+a \left (9+\frac {7 a^2}{b^2}\right ) x^2}{x^2 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac {b \sec ^4(c+d x) \left (3 a^2+b^2+a \left (3+\frac {a^2}{b^2}\right ) b \sin (c+d x)\right )}{4 d}+\frac {a b \sec ^2(c+d x) \left (12 a+\left (9+\frac {7 a^2}{b^2}\right ) b \sin (c+d x)\right )}{8 d}+\frac {b^3 \operatorname {Subst}\left (\int \left (\frac {3 a (a+b) (5 a+3 b)}{2 b^3 (b-x)}+\frac {8 a^3}{b^2 x^2}+\frac {24 a^2}{b^2 x}+\frac {3 a (5 a-3 b) (a-b)}{2 b^3 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=-\frac {a^3 \csc (c+d x)}{d}-\frac {3 a (a+b) (5 a+3 b) \log (1-\sin (c+d x))}{16 d}+\frac {3 a^2 b \log (\sin (c+d x))}{d}+\frac {3 a (5 a-3 b) (a-b) \log (1+\sin (c+d x))}{16 d}+\frac {b \sec ^4(c+d x) \left (3 a^2+b^2+a \left (3+\frac {a^2}{b^2}\right ) b \sin (c+d x)\right )}{4 d}+\frac {a b \sec ^2(c+d x) \left (12 a+\left (9+\frac {7 a^2}{b^2}\right ) b \sin (c+d x)\right )}{8 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.45, size = 161, normalized size = 0.94 \[ -\frac {16 a^3 \csc (c+d x)-48 a^2 b \log (\sin (c+d x))+\frac {(a+b)^2 (7 a+b)}{\sin (c+d x)-1}+\frac {(a-b)^2 (7 a-b)}{\sin (c+d x)+1}-\frac {(a+b)^3}{(\sin (c+d x)-1)^2}+\frac {(a-b)^3}{(\sin (c+d x)+1)^2}+3 a (a+b) (5 a+3 b) \log (1-\sin (c+d x))-3 a (5 a-3 b) (a-b) \log (\sin (c+d x)+1)}{16 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.47, size = 226, normalized size = 1.32 \[ \frac {48 \, a^{2} b \cos \left (d x + c\right )^{4} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 3 \, {\left (5 \, a^{3} - 8 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) - 3 \, {\left (5 \, a^{3} + 8 \, a^{2} b + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) - 6 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + 4 \, a^{3} + 12 \, a b^{2} + 2 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (6 \, a^{2} b \cos \left (d x + c\right )^{2} + 3 \, a^{2} b + b^{3}\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.30, size = 210, normalized size = 1.23 \[ \frac {48 \, a^{2} b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 3 \, {\left (5 \, a^{3} - 8 \, a^{2} b + 3 \, a b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 3 \, {\left (5 \, a^{3} + 8 \, a^{2} b + 3 \, a b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {16 \, {\left (3 \, a^{2} b \sin \left (d x + c\right ) + a^{3}\right )}}{\sin \left (d x + c\right )} + \frac {2 \, {\left (18 \, a^{2} b \sin \left (d x + c\right )^{4} - 7 \, a^{3} \sin \left (d x + c\right )^{3} - 9 \, a b^{2} \sin \left (d x + c\right )^{3} - 48 \, a^{2} b \sin \left (d x + c\right )^{2} + 9 \, a^{3} \sin \left (d x + c\right ) + 15 \, a b^{2} \sin \left (d x + c\right ) + 36 \, a^{2} b + 2 \, b^{3}\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.64, size = 221, normalized size = 1.29 \[ \frac {a^{3}}{4 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}+\frac {5 a^{3}}{8 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}-\frac {15 a^{3}}{8 d \sin \left (d x +c \right )}+\frac {15 a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {3 a^{2} b}{4 d \cos \left (d x +c \right )^{4}}+\frac {3 a^{2} b}{2 d \cos \left (d x +c \right )^{2}}+\frac {3 a^{2} b \ln \left (\tan \left (d x +c \right )\right )}{d}+\frac {3 a \,b^{2} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {9 a \,b^{2} \tan \left (d x +c \right ) \sec \left (d x +c \right )}{8 d}+\frac {9 a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {b^{3}}{4 d \cos \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 188, normalized size = 1.10 \[ \frac {48 \, a^{2} b \log \left (\sin \left (d x + c\right )\right ) + 3 \, {\left (5 \, a^{3} - 8 \, a^{2} b + 3 \, a b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (5 \, a^{3} + 8 \, a^{2} b + 3 \, a b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (12 \, a^{2} b \sin \left (d x + c\right )^{3} + 3 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (d x + c\right )^{4} + 8 \, a^{3} - 5 \, {\left (5 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (d x + c\right )^{2} - 2 \, {\left (9 \, a^{2} b + b^{3}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{5} - 2 \, \sin \left (d x + c\right )^{3} + \sin \left (d x + c\right )}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.13, size = 182, normalized size = 1.06 \[ \frac {3\,a^2\,b\,\ln \left (\sin \left (c+d\,x\right )\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^4\,\left (\frac {15\,a^3}{8}+\frac {9\,a\,b^2}{8}\right )-{\sin \left (c+d\,x\right )}^2\,\left (\frac {25\,a^3}{8}+\frac {15\,a\,b^2}{8}\right )+a^3-\sin \left (c+d\,x\right )\,\left (\frac {9\,a^2\,b}{4}+\frac {b^3}{4}\right )+\frac {3\,a^2\,b\,{\sin \left (c+d\,x\right )}^3}{2}}{d\,\left ({\sin \left (c+d\,x\right )}^5-2\,{\sin \left (c+d\,x\right )}^3+\sin \left (c+d\,x\right )\right )}+\frac {3\,a\,\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (a-b\right )\,\left (5\,a-3\,b\right )}{16\,d}-\frac {3\,a\,\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (a+b\right )\,\left (5\,a+3\,b\right )}{16\,d} \]
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]
________________________________________________________________________________________