Optimal. Leaf size=322 \[ -\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {c \sec (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right ) \sqrt {c \sec (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {\sqrt {c \sec (a+b x)} \log \left (\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{8 \sqrt {2} b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}-\frac {\sqrt {c \sec (a+b x)} \log \left (\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{8 \sqrt {2} b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}} \]
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Rubi [A] time = 0.21, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2628, 2629, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {c \sec (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right ) \sqrt {c \sec (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {\sqrt {c \sec (a+b x)} \log \left (\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{8 \sqrt {2} b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}-\frac {\sqrt {c \sec (a+b x)} \log \left (\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{8 \sqrt {2} b c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}+\frac {d}{2 b c \sqrt {c \sec (a+b x)} (d \csc (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 297
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2628
Rule 2629
Rule 3476
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d \csc (a+b x)} (c \sec (a+b x))^{3/2}} \, dx &=\frac {d}{2 b c (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}+\frac {\int \frac {\sqrt {c \sec (a+b x)}}{\sqrt {d \csc (a+b x)}} \, dx}{4 c^2}\\ &=\frac {d}{2 b c (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}+\frac {\sqrt {c \sec (a+b x)} \int \sqrt {\tan (a+b x)} \, dx}{4 c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}\\ &=\frac {d}{2 b c (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}+\frac {\sqrt {c \sec (a+b x)} \operatorname {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (a+b x)\right )}{4 b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}\\ &=\frac {d}{2 b c (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}+\frac {\sqrt {c \sec (a+b x)} \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (a+b x)}\right )}{2 b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}\\ &=\frac {d}{2 b c (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}-\frac {\sqrt {c \sec (a+b x)} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (a+b x)}\right )}{4 b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}+\frac {\sqrt {c \sec (a+b x)} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (a+b x)}\right )}{4 b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}\\ &=\frac {d}{2 b c (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}+\frac {\sqrt {c \sec (a+b x)} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (a+b x)}\right )}{8 b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}+\frac {\sqrt {c \sec (a+b x)} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (a+b x)}\right )}{8 b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}+\frac {\sqrt {c \sec (a+b x)} \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (a+b x)}\right )}{8 \sqrt {2} b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}+\frac {\sqrt {c \sec (a+b x)} \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (a+b x)}\right )}{8 \sqrt {2} b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}\\ &=\frac {d}{2 b c (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {c \sec (a+b x)}}{8 \sqrt {2} b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {c \sec (a+b x)}}{8 \sqrt {2} b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}+\frac {\sqrt {c \sec (a+b x)} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (a+b x)}\right )}{4 \sqrt {2} b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}-\frac {\sqrt {c \sec (a+b x)} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (a+b x)}\right )}{4 \sqrt {2} b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}\\ &=\frac {d}{2 b c (d \csc (a+b x))^{3/2} \sqrt {c \sec (a+b x)}}-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {c \sec (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}+\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {c \sec (a+b x)}}{4 \sqrt {2} b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {c \sec (a+b x)}}{8 \sqrt {2} b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (a+b x)}+\tan (a+b x)\right ) \sqrt {c \sec (a+b x)}}{8 \sqrt {2} b c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 2.02, size = 223, normalized size = 0.69 \[ \frac {\sqrt {d \csc (a+b x)} \left (4 \sqrt [4]{\cot ^2(a+b x)}-4 \cos (2 (a+b x)) \sqrt [4]{\cot ^2(a+b x)}+\sqrt {2} \log \left (\sqrt {\cot ^2(a+b x)}-\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}+1\right )-\sqrt {2} \log \left (\sqrt {\cot ^2(a+b x)}+\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}+1\right )+2 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}\right )-2 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt [4]{\cot ^2(a+b x)}+1\right )\right )}{16 b c d \sqrt [4]{\cot ^2(a+b x)} \sqrt {c \sec (a+b x)}} \]
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 \[ \int \frac {1}{\sqrt {d \csc \left (b x + a\right )} \left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.99, size = 520, normalized size = 1.61 \[ -\frac {\left (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 )-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 )-\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 )-\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 )-2 \sqrt {2}\, \left (\cos ^{2}\left (b x +a \right )\right )+2 \cos \left (b x +a \right ) \sqrt {2}\right ) \sin \left (b x +a \right ) \sqrt {2}}{8 b \left (-1+\cos \left (b x +a \right )\right ) \cos \left (b x +a \right )^{2} \sqrt {\frac {d}{\sin \left (b x +a \right )}}\, \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{\sqrt {d \csc \left (b x + a\right )} \left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{3/2}\,\sqrt {\frac {d}{\sin \left (a+b\,x\right )}}} \,d x \]
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 \[ \int \frac {1}{\left (c \sec {\left (a + b x \right )}\right )^{\frac {3}{2}} \sqrt {d \csc {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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