Optimal. Leaf size=327 \[ \frac {c^2 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}{\sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}-\frac {c^2 \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right ) \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}{\sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}+\frac {c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \log \left (\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}-\frac {c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \log \left (\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}+\frac {2 c \sqrt {c \sec (a+b x)}}{b d \sqrt {d \csc (a+b x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.22, antiderivative size = 327, 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 = {2624, 2629, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac {c^2 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}{\sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}-\frac {c^2 \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (a+b x)}+1\right ) \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)}}{\sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}+\frac {c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \log \left (\tan (a+b x)-\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}-\frac {c^2 \sqrt {\tan (a+b x)} \sqrt {d \csc (a+b x)} \log \left (\tan (a+b x)+\sqrt {2} \sqrt {\tan (a+b x)}+1\right )}{2 \sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}+\frac {2 c \sqrt {c \sec (a+b x)}}{b d \sqrt {d \csc (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 211
Rule 329
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 2624
Rule 2629
Rule 3476
Rubi steps
\begin {align*} \int \frac {(c \sec (a+b x))^{3/2}}{(d \csc (a+b x))^{3/2}} \, dx &=\frac {2 c \sqrt {c \sec (a+b x)}}{b d \sqrt {d \csc (a+b x)}}-\frac {c^2 \int \frac {\sqrt {d \csc (a+b x)}}{\sqrt {c \sec (a+b x)}} \, dx}{d^2}\\ &=\frac {2 c \sqrt {c \sec (a+b x)}}{b d \sqrt {d \csc (a+b x)}}-\frac {\left (c^2 \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}\right ) \int \frac {1}{\sqrt {\tan (a+b x)}} \, dx}{d^2 \sqrt {c \sec (a+b x)}}\\ &=\frac {2 c \sqrt {c \sec (a+b x)}}{b d \sqrt {d \csc (a+b x)}}-\frac {\left (c^2 \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 )}{b d^2 \sqrt {c \sec (a+b x)}}\\ &=\frac {2 c \sqrt {c \sec (a+b x)}}{b d \sqrt {d \csc (a+b x)}}-\frac {\left (2 c^2 \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 )}{b d^2 \sqrt {c \sec (a+b x)}}\\ &=\frac {2 c \sqrt {c \sec (a+b x)}}{b d \sqrt {d \csc (a+b x)}}-\frac {\left (c^2 \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 )}{b d^2 \sqrt {c \sec (a+b x)}}-\frac {\left (c^2 \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 )}{b d^2 \sqrt {c \sec (a+b x)}}\\ &=\frac {2 c \sqrt {c \sec (a+b x)}}{b d \sqrt {d \csc (a+b x)}}-\frac {\left (c^2 \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 )}{2 b d^2 \sqrt {c \sec (a+b x)}}-\frac {\left (c^2 \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 )}{2 b d^2 \sqrt {c \sec (a+b x)}}+\frac {\left (c^2 \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 )}{2 \sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}+\frac {\left (c^2 \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 )}{2 \sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}\\ &=\frac {2 c \sqrt {c \sec (a+b x)}}{b d \sqrt {d \csc (a+b x)}}+\frac {c^2 \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)}}{2 \sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}-\frac {c^2 \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)}}{2 \sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}-\frac {\left (c^2 \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 )}{\sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}+\frac {\left (c^2 \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 )}{\sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}\\ &=\frac {2 c \sqrt {c \sec (a+b x)}}{b d \sqrt {d \csc (a+b x)}}+\frac {c^2 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{\sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}-\frac {c^2 \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (a+b x)}\right ) \sqrt {d \csc (a+b x)} \sqrt {\tan (a+b x)}}{\sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}+\frac {c^2 \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)}}{2 \sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}-\frac {c^2 \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)}}{2 \sqrt {2} b d^2 \sqrt {c \sec (a+b x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.25, size = 64, normalized size = 0.20 \[ \frac {2 c \sqrt {c \sec (a+b x)} \left (\cot ^2(a+b x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(a+b x)\right )+3\right )}{3 b d \sqrt {d \csc (a+b x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}}{\left (d \csc \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 1.06, size = 646, normalized size = 1.98 \[ \frac {\left (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 )-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 )+\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 )+\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 \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 )+2 \cos \left (b x +a \right ) \sqrt {2}-2 \sqrt {2}\right ) \cos \left (b x +a \right ) \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {3}{2}} \sqrt {2}}{2 b \left (-1+\cos \left (b x +a \right )\right ) \left (\frac {d}{\sin \left (b x +a \right )}\right )^{\frac {3}{2}} \sin \left (b x +a \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \sec \left (b x + a\right )\right )^{\frac {3}{2}}}{\left (d \csc \left (b x + a\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{3/2}}{{\left (\frac {d}{\sin \left (a+b\,x\right )}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________