Optimal. Leaf size=371 \[ -\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}{32 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}+\frac {3 \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right ) \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}{32 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}-\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \log \left (\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{64 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}+\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \log \left (\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{64 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}+\frac {1}{16 b c d (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}} \]
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Rubi [A] time = 0.29, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2627, 2628, 2629, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}{32 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}+\frac {3 \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right ) \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}{32 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}-\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \log \left (\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{64 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}+\frac {3 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \log \left (\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{64 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}-\frac {c}{4 b d (c \sec (a+b x))^{7/2} \sqrt {d \csc (a+b x)}}+\frac {1}{16 b c d (c \sec (a+b x))^{3/2} \sqrt {d \csc (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2627
Rule 2628
Rule 2629
Rule 3476
Rubi steps
\begin {align*} \int \frac {1}{(d \csc (a+b x))^{3/2} (c \sec (a+b x))^{5/2}} \, dx &=-\frac {c}{4 b d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac {\int \frac {\sqrt {d \csc (a+b x)}}{(c \sec (a+b x))^{5/2}} \, dx}{8 d^2}\\ &=-\frac {c}{4 b d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac {1}{16 b c d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac {3 \int \frac {\sqrt {d \csc (a+b x)}}{\sqrt {c \sec (a+b x)}} \, dx}{32 c^2 d^2}\\ &=-\frac {c}{4 b d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac {1}{16 b c d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \int \frac {1}{\sqrt {\tan (a+b x)}} \, dx}{32 c^2 d^2 \sqrt {c \sec (a+b x)}}\\ &=-\frac {c}{4 b d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac {1}{16 b c d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (a+b x)\right )}{32 b c^2 d^2 \sqrt {c \sec (a+b x)}}\\ &=-\frac {c}{4 b d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac {1}{16 b c d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\tan (a+b x)}\right )}{16 b c^2 d^2 \sqrt {c \sec (a+b x)}}\\ &=-\frac {c}{4 b d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac {1}{16 b c d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (a+b x)}\right )}{32 b c^2 d^2 \sqrt {c \sec (a+b x)}}+\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (a+b x)}\right )}{32 b c^2 d^2 \sqrt {c \sec (a+b x)}}\\ &=-\frac {c}{4 b d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac {1}{16 b c d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}+\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (a+b x)}\right )}{64 b c^2 d^2 \sqrt {c \sec (a+b x)}}+\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (a+b x)}\right )}{64 b c^2 d^2 \sqrt {c \sec (a+b x)}}-\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (a+b x)}\right )}{64 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}-\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (a+b x)}\right )}{64 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}\\ &=-\frac {c}{4 b d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac {1}{16 b c d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}-\frac {3 \sqrt {d \csc (a+b x)} \log \left (1-\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {\tan (a+b x)}}{64 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}+\frac {3 \sqrt {d \csc (a+b x)} \log \left (1+\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {\tan (a+b x)}}{64 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}+\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{32 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}-\frac {\left (3 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (a+b x)}\right )}{32 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}\\ &=-\frac {c}{4 b d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{7/2}}+\frac {1}{16 b c d \sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{32 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}+\frac {3 \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{32 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}-\frac {3 \sqrt {d \csc (a+b x)} \log \left (1-\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {\tan (a+b x)}}{64 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}+\frac {3 \sqrt {d \csc (a+b x)} \log \left (1+\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {\tan (a+b x)}}{64 \sqrt {2} b c^2 d^2 \sqrt {c \sec (a+b x)}}\\ \end {align*}
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Mathematica [C] time = 0.37, size = 81, normalized size = 0.22 \[ -\frac {\csc ^3(a+b x) \left (2 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(a+b x)\right )+\cos (2 (a+b x))-\cos (4 (a+b x))\right )}{32 b c (c \sec (a+b x))^{3/2} (d \csc (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d \csc \left (b x + a\right )\right )^{\frac {3}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.86, size = 686, normalized size = 1.85 \[ -\frac {\left (3 i \EllipticPi \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right )-3 i \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sin \left (b x +a \right )+8 \left (\cos ^{5}\left (b x +a \right )\right ) \sqrt {2}-8 \left (\cos ^{4}\left (b x +a \right )\right ) \sqrt {2}-6 \sin \left (b x +a \right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {\sqrt {2}}{2}\right )+3 \EllipticPi \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sin \left (b x +a \right )+3 \sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \sqrt {\frac {-1+\cos \left (b x +a \right )}{\sin \left (b x +a \right )}}\, \EllipticPi \left (\sqrt {\frac {1-\cos \left (b x +a \right )+\sin \left (b x +a \right )}{\sin \left (b x +a \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sin \left (b x +a \right )-2 \left (\cos ^{3}\left (b x +a \right )\right ) \sqrt {2}+2 \sqrt {2}\, \left (\cos ^{2}\left (b x +a \right )\right )\right ) \sqrt {2}}{64 b \left (-1+\cos \left (b x +a \right )\right ) \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {3}{2}} \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {5}{2}} \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d \csc \left (b x + a\right )\right )^{\frac {3}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{5/2}\,{\left (\frac {d}{\sin \left (a+b\,x\right )}\right )}^{3/2}} \,d x \]
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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