Optimal. Leaf size=189 \[ -\frac {\left (a^2+3 b^2\right ) \sin (c+d x)}{d}-\frac {\left (15 a^2+48 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac {\left (15 a^2-48 a b+35 b^2\right ) \log (\sin (c+d x)+1)}{16 d}-\frac {a b \sin ^2(c+d x)}{d}-\frac {\sec ^2(c+d x) (9 a \sin (c+d x)+11 b) (a+b \sin (c+d x))}{8 d}+\frac {\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {b^2 \sin ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.36, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2837, 12, 1645, 1810, 633, 31} \[ -\frac {\left (a^2+3 b^2\right ) \sin (c+d x)}{d}-\frac {\left (15 a^2+48 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac {\left (15 a^2-48 a b+35 b^2\right ) \log (\sin (c+d x)+1)}{16 d}-\frac {a b \sin ^2(c+d x)}{d}-\frac {\sec ^2(c+d x) (9 a \sin (c+d x)+11 b) (a+b \sin (c+d x))}{8 d}+\frac {\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {b^2 \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 633
Rule 1645
Rule 1810
Rule 2837
Rubi steps
\begin {align*} \int \sin (c+d x) (a+b \sin (c+d x))^2 \tan ^5(c+d x) \, dx &=\frac {b^5 \operatorname {Subst}\left (\int \frac {x^6 (a+x)^2}{b^6 \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^6 (a+x)^2}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^2 \tan (c+d x)}{4 d}+\frac {\operatorname {Subst}\left (\int \frac {(a+x) \left (-a b^6-3 b^6 x-4 a b^4 x^2-4 b^4 x^3-4 a b^2 x^4-4 b^2 x^5\right )}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 b^3 d}\\ &=-\frac {\sec ^2(c+d x) (11 b+9 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))^2 \tan (c+d x)}{4 d}+\frac {\operatorname {Subst}\left (\int \frac {b^6 \left (7 a^2+11 b^2\right )+32 a b^6 x+8 b^4 \left (a^2+2 b^2\right ) x^2+16 a b^4 x^3+8 b^4 x^4}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 b^5 d}\\ &=-\frac {\sec ^2(c+d x) (11 b+9 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))^2 \tan (c+d x)}{4 d}+\frac {\operatorname {Subst}\left (\int \left (-8 b^4 \left (a^2+3 b^2\right )-16 a b^4 x-8 b^4 x^2+\frac {5 b^6 \left (3 a^2+7 b^2\right )+48 a b^6 x}{b^2-x^2}\right ) \, dx,x,b \sin (c+d x)\right )}{8 b^5 d}\\ &=-\frac {\left (a^2+3 b^2\right ) \sin (c+d x)}{d}-\frac {a b \sin ^2(c+d x)}{d}-\frac {b^2 \sin ^3(c+d x)}{3 d}-\frac {\sec ^2(c+d x) (11 b+9 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))^2 \tan (c+d x)}{4 d}+\frac {\operatorname {Subst}\left (\int \frac {5 b^6 \left (3 a^2+7 b^2\right )+48 a b^6 x}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 b^5 d}\\ &=-\frac {\left (a^2+3 b^2\right ) \sin (c+d x)}{d}-\frac {a b \sin ^2(c+d x)}{d}-\frac {b^2 \sin ^3(c+d x)}{3 d}-\frac {\sec ^2(c+d x) (11 b+9 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))^2 \tan (c+d x)}{4 d}-\frac {\left (15 a^2-48 a b+35 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{16 d}+\frac {\left (15 a^2+48 a b+35 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{b-x} \, dx,x,b \sin (c+d x)\right )}{16 d}\\ &=-\frac {\left (15 a^2+48 a b+35 b^2\right ) \log (1-\sin (c+d x))}{16 d}+\frac {\left (15 a^2-48 a b+35 b^2\right ) \log (1+\sin (c+d x))}{16 d}-\frac {\left (a^2+3 b^2\right ) \sin (c+d x)}{d}-\frac {a b \sin ^2(c+d x)}{d}-\frac {b^2 \sin ^3(c+d x)}{3 d}-\frac {\sec ^2(c+d x) (11 b+9 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))^2 \tan (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 1.56, size = 186, normalized size = 0.98 \[ \frac {-48 \left (a^2+3 b^2\right ) \sin (c+d x)-3 \left (15 a^2+48 a b+35 b^2\right ) \log (1-\sin (c+d x))+3 \left (15 a^2-48 a b+35 b^2\right ) \log (\sin (c+d x)+1)-48 a b \sin ^2(c+d x)+\frac {3 (a+b) (9 a+13 b)}{\sin (c+d x)-1}+\frac {3 (9 a-13 b) (a-b)}{\sin (c+d x)+1}+\frac {3 (a+b)^2}{(\sin (c+d x)-1)^2}-\frac {3 (a-b)^2}{(\sin (c+d x)+1)^2}-16 b^2 \sin ^3(c+d x)}{48 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 198, normalized size = 1.05 \[ \frac {48 \, a b \cos \left (d x + c\right )^{6} - 24 \, a b \cos \left (d x + c\right )^{4} + 3 \, {\left (15 \, a^{2} - 48 \, a b + 35 \, b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (15 \, a^{2} + 48 \, a b + 35 \, b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 144 \, a b \cos \left (d x + c\right )^{2} + 24 \, a b + 2 \, {\left (8 \, b^{2} \cos \left (d x + c\right )^{6} - 8 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 3 \, {\left (9 \, a^{2} + 13 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 6 \, a^{2} + 6 \, b^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 198, normalized size = 1.05 \[ -\frac {16 \, b^{2} \sin \left (d x + c\right )^{3} + 48 \, a b \sin \left (d x + c\right )^{2} + 48 \, a^{2} \sin \left (d x + c\right ) + 144 \, b^{2} \sin \left (d x + c\right ) - 3 \, {\left (15 \, a^{2} - 48 \, a b + 35 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 3 \, {\left (15 \, a^{2} + 48 \, a b + 35 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {6 \, {\left (36 \, a b \sin \left (d x + c\right )^{4} + 9 \, a^{2} \sin \left (d x + c\right )^{3} + 13 \, b^{2} \sin \left (d x + c\right )^{3} - 48 \, a b \sin \left (d x + c\right )^{2} - 7 \, a^{2} \sin \left (d x + c\right ) - 11 \, b^{2} \sin \left (d x + c\right ) + 16 \, a b\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 355, normalized size = 1.88 \[ \frac {a^{2} \left (\sin ^{7}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {3 a^{2} \left (\sin ^{7}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}-\frac {3 a^{2} \left (\sin ^{5}\left (d x +c \right )\right )}{8 d}-\frac {5 a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{8 d}-\frac {15 a^{2} \sin \left (d x +c \right )}{8 d}+\frac {15 a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {a b \left (\sin ^{8}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{4}}-\frac {a b \left (\sin ^{8}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}-\frac {a b \left (\sin ^{6}\left (d x +c \right )\right )}{d}-\frac {3 a b \left (\sin ^{4}\left (d x +c \right )\right )}{2 d}-\frac {3 a b \left (\sin ^{2}\left (d x +c \right )\right )}{d}-\frac {6 a b \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {b^{2} \left (\sin ^{9}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {5 b^{2} \left (\sin ^{9}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}-\frac {5 b^{2} \left (\sin ^{7}\left (d x +c \right )\right )}{8 d}-\frac {7 b^{2} \left (\sin ^{5}\left (d x +c \right )\right )}{8 d}-\frac {35 b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{24 d}-\frac {35 b^{2} \sin \left (d x +c \right )}{8 d}+\frac {35 b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 180, normalized size = 0.95 \[ -\frac {16 \, b^{2} \sin \left (d x + c\right )^{3} + 48 \, a b \sin \left (d x + c\right )^{2} - 3 \, {\left (15 \, a^{2} - 48 \, a b + 35 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (15 \, a^{2} + 48 \, a b + 35 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + 48 \, {\left (a^{2} + 3 \, b^{2}\right )} \sin \left (d x + c\right ) - \frac {6 \, {\left (24 \, a b \sin \left (d x + c\right )^{2} + {\left (9 \, a^{2} + 13 \, b^{2}\right )} \sin \left (d x + c\right )^{3} - 20 \, a b - {\left (7 \, a^{2} + 11 \, b^{2}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.33, size = 433, normalized size = 2.29 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,\left (\frac {15\,a^2}{8}-6\,a\,b+\frac {35\,b^2}{8}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,\left (\frac {15\,a^2}{8}+6\,a\,b+\frac {35\,b^2}{8}\right )}{d}-\frac {\left (-\frac {15\,a^2}{4}-\frac {35\,b^2}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-12\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\left (\frac {5\,a^2}{2}+\frac {35\,b^2}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+12\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\left (\frac {47\,a^2}{4}+\frac {329\,b^2}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+32\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\left (11\,a^2-17\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+32\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {47\,a^2}{4}+\frac {329\,b^2}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+12\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (\frac {5\,a^2}{2}+\frac {35\,b^2}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-12\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\left (-\frac {15\,a^2}{4}-\frac {35\,b^2}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}+\frac {6\,a\,b\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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