Optimal. Leaf size=94 \[ \frac {a^2 \log \left (a+b \sin ^2(c+d x)\right )}{2 d (a+b)^3}-\frac {a^2 \log (\cos (c+d x))}{d (a+b)^3}+\frac {\sec ^4(c+d x)}{4 d (a+b)}-\frac {(2 a+b) \sec ^2(c+d x)}{2 d (a+b)^2} \]
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Rubi [A] time = 0.10, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3194, 88} \[ \frac {a^2 \log \left (a+b \sin ^2(c+d x)\right )}{2 d (a+b)^3}-\frac {a^2 \log (\cos (c+d x))}{d (a+b)^3}+\frac {\sec ^4(c+d x)}{4 d (a+b)}-\frac {(2 a+b) \sec ^2(c+d x)}{2 d (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 88
Rule 3194
Rubi steps
\begin {align*} \int \frac {\tan ^5(c+d x)}{a+b \sin ^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{(1-x)^3 (a+b x)} \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{(a+b) (-1+x)^3}+\frac {-2 a-b}{(a+b)^2 (-1+x)^2}-\frac {a^2}{(a+b)^3 (-1+x)}+\frac {a^2 b}{(a+b)^3 (a+b x)}\right ) \, dx,x,\sin ^2(c+d x)\right )}{2 d}\\ &=-\frac {a^2 \log (\cos (c+d x))}{(a+b)^3 d}+\frac {a^2 \log \left (a+b \sin ^2(c+d x)\right )}{2 (a+b)^3 d}-\frac {(2 a+b) \sec ^2(c+d x)}{2 (a+b)^2 d}+\frac {\sec ^4(c+d x)}{4 (a+b) d}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 78, normalized size = 0.83 \[ \frac {-2 \left (2 a^2+3 a b+b^2\right ) \sec ^2(c+d x)+2 a^2 \left (\log \left (a+b \sin ^2(c+d x)\right )-2 \log (\cos (c+d x))\right )+(a+b)^2 \sec ^4(c+d x)}{4 d (a+b)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 118, normalized size = 1.26 \[ \frac {2 \, a^{2} \cos \left (d x + c\right )^{4} \log \left (-b \cos \left (d x + c\right )^{2} + a + b\right ) - 4 \, a^{2} \cos \left (d x + c\right )^{4} \log \left (-\cos \left (d x + c\right )\right ) - 2 \, {\left (2 \, a^{2} + 3 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}{4 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.75, size = 393, normalized size = 4.18 \[ \frac {\frac {6 \, a^{2} \log \left (a - \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {4 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {12 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {25 \, a^{2} + \frac {124 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {24 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {246 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {144 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {48 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {124 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {24 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {25 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 109, normalized size = 1.16 \[ \frac {a^{2} \ln \left (b \left (\cos ^{2}\left (d x +c \right )\right )-a -b \right )}{2 d \left (a +b \right )^{3}}-\frac {a}{d \left (a +b \right )^{2} \cos \left (d x +c \right )^{2}}-\frac {b}{2 d \left (a +b \right )^{2} \cos \left (d x +c \right )^{2}}+\frac {1}{4 d \left (a +b \right ) \cos \left (d x +c \right )^{4}}-\frac {a^{2} \ln \left (\cos \left (d x +c \right )\right )}{\left (a +b \right )^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 159, normalized size = 1.69 \[ \frac {\frac {2 \, a^{2} \log \left (b \sin \left (d x + c\right )^{2} + a\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {2 \, a^{2} \log \left (\sin \left (d x + c\right )^{2} - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} + \frac {2 \, {\left (2 \, a + b\right )} \sin \left (d x + c\right )^{2} - 3 \, a - b}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sin \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sin \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.35, size = 90, normalized size = 0.96 \[ \frac {a^2\,\left (\frac {\ln \left (\left (a+b\right )\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\right )}{2}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{2}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4}{4}\right )+\frac {b^2\,{\mathrm {tan}\left (c+d\,x\right )}^4}{4}-a\,b\,\left (\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{2}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^4}{2}\right )}{d\,{\left (a+b\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{5}{\left (c + d x \right )}}{a + b \sin ^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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