Optimal. Leaf size=155 \[ \frac {a \cos ^2(c+d x)}{2 d}+\frac {a \sec ^4(c+d x)}{4 d}-\frac {3 a \sec ^2(c+d x)}{2 d}-\frac {3 a \log (\cos (c+d x))}{d}-\frac {35 b \sin ^3(c+d x)}{24 d}-\frac {35 b \sin (c+d x)}{8 d}+\frac {b \sin ^3(c+d x) \tan ^4(c+d x)}{4 d}-\frac {7 b \sin ^3(c+d x) \tan ^2(c+d x)}{8 d}+\frac {35 b \tanh ^{-1}(\sin (c+d x))}{8 d} \]
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Rubi [A] time = 0.17, antiderivative size = 155, normalized size of antiderivative = 1.00, 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.
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Rule 43
Rule 206
Rule 266
Rule 288
Rule 302
Rule 2590
Rule 2592
Rule 2834
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*}
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Mathematica [A] time = 0.60, 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.
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fricas [A] time = 0.45, size = 149, normalized size = 0.96 \[ \frac {24 \, a \cos \left (d x + c\right )^{6} - 3 \, {\left (24 \, a - 35 \, b\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (24 \, a + 35 \, b\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 12 \, a \cos \left (d x + c\right )^{4} - 72 \, a \cos \left (d x + c\right )^{2} + 2 \, {\left (8 \, b \cos \left (d x + c\right )^{6} - 80 \, b \cos \left (d x + c\right )^{4} - 39 \, b \cos \left (d x + c\right )^{2} + 6 \, b\right )} \sin \left (d x + c\right ) + 12 \, a}{48 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 135, normalized size = 0.87 \[ -\frac {16 \, b \sin \left (d x + c\right )^{3} + 24 \, a \sin \left (d x + c\right )^{2} + 3 \, {\left (24 \, a - 35 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 3 \, {\left (24 \, a + 35 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + 144 \, b \sin \left (d x + c\right ) - \frac {6 \, {\left (18 \, a \sin \left (d x + c\right )^{4} + 13 \, b \sin \left (d x + c\right )^{3} - 24 \, a \sin \left (d x + c\right )^{2} - 11 \, b \sin \left (d x + c\right ) + 8 \, a\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 219, normalized size = 1.41 \[ \frac {a \left (\sin ^{8}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {a \left (\sin ^{8}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}-\frac {a \left (\sin ^{6}\left (d x +c \right )\right )}{2 d}-\frac {3 a \left (\sin ^{4}\left (d x +c \right )\right )}{4 d}-\frac {3 a \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}-\frac {3 a \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {b \left (\sin ^{9}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {5 b \left (\sin ^{9}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}-\frac {5 b \left (\sin ^{7}\left (d x +c \right )\right )}{8 d}-\frac {7 b \left (\sin ^{5}\left (d x +c \right )\right )}{8 d}-\frac {35 b \left (\sin ^{3}\left (d x +c \right )\right )}{24 d}-\frac {35 b \sin \left (d x +c \right )}{8 d}+\frac {35 b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 132, normalized size = 0.85 \[ -\frac {16 \, b \sin \left (d x + c\right )^{3} + 24 \, a \sin \left (d x + c\right )^{2} + 3 \, {\left (24 \, a - 35 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (24 \, a + 35 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + 144 \, b \sin \left (d x + c\right ) - \frac {6 \, {\left (13 \, b \sin \left (d x + c\right )^{3} + 12 \, a \sin \left (d x + c\right )^{2} - 11 \, b \sin \left (d x + c\right ) - 10 \, a\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.27, size = 346, normalized size = 2.23 \[ \frac {3\,a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {-\frac {35\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\frac {35\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{6}+6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {329\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}+16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-17\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+16\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {329\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {35\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6}-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {35\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,\left (3\,a+\frac {35\,b}{8}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,\left (3\,a-\frac {35\,b}{8}\right )}{d} \]
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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